text:
\section{Introduction}
Let $\alpha_1, \dots, \alpha_{N-1}$ be positive integers with
$\alpha_1, \dots, \alpha_{N-1} \geq 2$ and
$w_j=(w^{1}_{j}, \dots, \\ w^{m_j}_{j}) \in \mathbb{C}^{m_j},
w=(w_1, \dots, w_N) \in \mathbb{C}^{m_1}
\times \dots \times \mathbb{C}^{m_N}$, $m_1+\dots+m_N=m$
and $z \in \mathbb{C}$.
Let $R(z, w)$ be a real analytic function defined by
\begin{equation}
R(z, w)=|z|^2+\sum^{N-1}_{j=1}||w_j||^{2\alpha_j}+||w_N||^2-1, \nonumber
\end{equation}
where $||w_j||^2=|w^{1}_{j}|^2+\dots+|w^{m_j}_{j}|^2$ denotes the squared Euclidean norm.

Let $E(m; m_1, \dots, m_N; \alpha_1, \dots, \alpha_{N-1}, 1)$
be a bounded domain
in $\mathbb{C}^{m+1}$ with real analytic boundary defined by
the following:
\begin{align}
&E(m; m_1, \dots, m_N; \alpha_1, \dots, \alpha_{N-1}, 1) \nn \\
&\qquad \qquad =\{(z, w_1, \dots, w_N) \in \mathbb{C} \times \mathbb{C}^{m_1}
\times \dots \times
\mathbb{C}^{m_N} : R(z, w)<0\}. \nonumber
\end{align}
We call this domain a generalized pseudoellipsoid with $N$ blocks.
We abbreviate
$E(m; m_1, \dots, m_N; \alpha_1, \dots, \alpha_{N-1}, 1)$ as
$E(m; (m); (\alpha))$.
We sometimes write $\alpha_N=1$.

Let $\Hm$ be a pseudoconvex domain defined by
\begin{align}
\Hm=&
\{(z, w_1, \dots, w_N) \in \mathbb{C}\times \mathbb{C}^{m_1} \times \dots
      \times \mathbb{C}^{m_N} : \nn \\
     &\text{Im}~z>||w_1||^{2\alpha_1}+\dots+||w_{N-1}||^{2\alpha_{N-1}}
         +||w_N||^2\}.   \nonumber
\end{align}
The generalized pseudoellipsoid $E(m; (m); (\alpha))$ is biholomorphically equivalent to
its unbounded representation $\Hm$
via the mapping
\begin{align}
\Psi & :  \Hm \ni (z, w_1, \dots, w_N)  \label{emhm}\\
& \quad \mapsto
(\dfrac{i-z}{i+z}, \dfrac{2^{1/\alpha_1}w_1}{(i+z)^{1/\alpha_1}}, \dots,
\dfrac{2^{1/\alpha_{N-1}}w_{N-1}}{(i+z)^{1/\alpha_{N-1}}},
\dfrac{2w_N}{i+z}) \in E(m; (m); (\alpha)).  \nonumber
\end{align}

We study a proper holomorphic mapping
\begin{equation}
(\mathcal{F}, \mathcal{G})=(\mathcal{F}, \mathcal{G}_1, \dots, \mathcal{G}_N):\Em \to \En \nn
\end{equation}
or its unbounded representation
\begin{equation}
\Phi=\wt{\Psi}^{-1} \circ (\mathcal{F}, \mathcal{G}) \circ \Psi : \Hm \to \Hn,  \nn
\end{equation}
here $\wt{\Psi}$ is a mapping between $\Hn$ and $\En$ defined analogously by  (\ref{emhm}).
Let $\Phi=(F, G)=(F_1, G_1, \dots, G_N)$.
If $(\mathcal{F}, \mathcal{G})$ extends holomorphically past the boundary, so does $(F, G)$.
In this case, we use the extended mapping $(F, G)$ in two meanings without any mention:
one is the mapping between their closures: $\ol{\Hm} \to \ol{\Hn}$
and the other is  its restriction to the boundaries: $\pa \Hm \to \pa \Hn$.

Let $f, g : D_1 \to D_2$ be mappings with the same source and target domains.
We say that $f$ is equivalent to $g$
if there exist automorphisms $\phi_1 \in \text{Aut}(D_1)$ and $\phi_2 \in \text{Aut}(D_2)$
such that $f=\phi_2 \circ g \circ \phi_1$ holds.

Now we introduce a homogeneous proper holomorphic mapping $H_M(z)$
between balls of different dimensions.
For a variable $z=(z_1, \dots, z_n) \in \mathbb{C}^n$, we define the tensor product by
\begin{equation}
z \otimes z=(z_1z_1,  z_2z_1, \dots, z_nz_1, z_1z_2, \dots, z_nz_n). \nn
\end{equation}
Namely, the tensor product of $z$ and $z$ is the mapping whose components
are all possible products of the components of $z$.
Repeat tensor products $M$ times and after applying a unitary transformation
that collects the same monomials together
and ignoring zero components, we denote by the resulted mapping $H_M(z)$,
which has an expression:
\begin{equation}
H_M(z)=(\dots, \sqrt{\dfrac{M!}{p_1! \dots p_n!}}z^p, \dots) \label{h}
\end{equation}
for $p=(p_1, \dots, p_n) \in (\mathbb{Z}_{\geq 0})^n, \; p_1+\dots+p_n=M$.
D'Angelo introduced this mapping to classify proper holomorphic mappings
between balls of different dimensions, in fact,
\begin{equation}
H_M(z) : B^n \to B^N,
N=\begin{pmatrix}
n+M-1 \\
M
\end{pmatrix} \nn
\end{equation}
is a proper holomorphic mapping and satisfies the equality
$||z||^{2M}=||H_M(z)||^2$.
For this mapping, refer the book \cite{D}.

With the above notations and terminology, we can state our result as follows:
\begin{main theorem}\label{main}
Let $\Em$ and $\En$ be generalized pseudoellipsoids with $N$ blocks.
Let $(\mathcal{F}, \mathcal{G}_1, \dots, \mathcal{G}_N) : \Em \to \En$
be a proper holomorphic mapping that is holomorphic up to the boundary.
Assume that all components $\mathcal{F}, \mathcal{G}_1, \dots, \mathcal{G}_N$
are not constant and that $m_j \geq 2, j=1, \dots, N-1$.
Then we have the following$\mathrm{:}$

$\mathrm{(1)}$ There exists a permutation $\sigma$
of $\{1, \dots, N-1\}$ such that
\begin{equation}
\mathcal{G}_j|_{w_{\sigma(j)}=0}=0, \quad j=1, \dots, N-1. \nn
\end{equation}

$\mathrm{(2)}$
There exist integers $M_1, \dots, M_{N-1}$ such that
$\alpha_{\sigma(j)}=M_j \beta_j, j=1, \dots, N-1$.

$\mathrm{(3)}$ $n_j \geq \dfrac{(M_j+m_{\sigma(j)}-1)!}{M_j ! (m_{\sigma(j)}-1)!}, \quad
 j=1, \dots, N-1$.

$\mathrm{(4)}$ $(\mathcal{F}, \mathcal{G}_1, \dots, \mathcal{G}_N)$ is equivalent to
$(\widetilde{\mathcal{F}}, \widetilde{\mathcal{G}}_1, \dots, \widetilde{\mathcal{G}}_N)$
of the form
\begin{align}
&\widetilde{\mathcal{F}}(z, w_1, \dots, w_N)=z, \nn \\
&\widetilde{\mathcal{G}}_j(z, w_1, \dots, w_N)=(H_{M_j}(w_{\sigma(j)}), 0),  \quad
    j=1, \dots, N-1, \nn \\
&\widetilde{\mathcal{G}}_N(z, w_1, \dots, w_N)=(w_N, 0). \nn
\end{align}
\end{main theorem}

\begin{remark}
Outline of the proof is the following.

$\mathrm{(1)}$ Decompose a CR vector bundle into a special kind of
direct sum and the decomposition derives
expansions of the mapping. See Sections~$\mathrm{2}$ and $\mathrm{3}$.

$\mathrm{(2)}$ Restricting our situation to a certain variety,
we get the relations of exponents
$\alpha_j$ and $\beta_j$. See Section~$\mathrm{4}$.

$\mathrm{(3)}$ Again we restrict to a certain variety to reduce
an $N$ blocks case to a one block case.
Then we obtain the normalization of a certain component of the mapping.
Varying the variety to restrict, we obtain the whole normalization of the mapping.
See Sections~$\mathrm{5}$ and $\mathrm{6}$.
\end{remark}
If $m_j=1$, we call the domain $E(m; (1), \alpha)$ a pseudoellipsoid.
For pseudoellipsoids and generalized pseudoellipsoids, following topics have been studied.

(1) Given two (generalized) pseudoellipsoids, find necessary and sufficient conditions
to exist proper holomorphic mappings between them.
Classify the mappings between them.
\cite{DP1}, \cite{DP2}, \cite{Ha1}, \cite{Ha2}, \cite{K4}, \cite{L}, \cite{MM}

(2) Given any local biholomorphic mapping on a neighborhood of a boundary point
such that the point is mapped to a boundary point,
prove that it can be extended to a global biholomorphic mapping.
\cite{DP2}, \cite{K3}

(3) Characterize (generalized) pseudoellipsoids by mean of their
holomorphic automorphism groups.
\cite{GK}, \cite{K1}, \cite{K2}, \cite{K3}, \cite{KKM}

Present article is related to (1).
The first result on this line is done by M.~Landucci \cite{L}.
He studied pseudoellipsoids
\begin{equation}
E(\alpha)=\{z \in \mathbb{C}^n : \sum^{n}_{k=1}|z_k|^{2\alpha_k}<1\} \nn
\end{equation}
for $\alpha_k \in \mathbb{N}$.
If there exists a proper holomorphic mapping $f:E(\alpha) \to E(\beta)$,
then, after reordering the variables,
$\alpha_j/\beta_j \in \mathbb{N}$ holds,
and the mapping is equivalent to
$(z_1, \dots, z_n) \mapsto (z_1^{\alpha_1/\beta_1}, \dots, z_n^{\alpha_n/\beta_n})$.
Another proof of the relations between $\alpha_j$ and $\beta_j$
was given by the author
\cite{Ha1} using the tangential Cauchy-Riemann equation.
After that, G.~Dini-A.~S.~Primicerio \cite{DP1} proved the case of
$\alpha_j \in \mathbb{R}_{>0}$ in Landucci's theorem.
The determination of the mappings between generalized pseudoellipsoids
was given by R.~Monti-D.~Morbidelli \cite{MM}.
They studied local CR mappings, which are viewed as boundary values of
proper holomorphic mappings,
between boundaries of generalized pseudoellipsoids
and decomposed them to elementary mappings, namely, any such local CR mappings are
composite mappings of inversions, one-parameter group of dilations,
a kind of unitary transformations, and shifts.
G.~Dini and A.~S.~Primicerio studied more general domain,
which is denoted by $\Sigma(m,\beta, d)$, and obtained the relations of indices.
Let $\Sigma(m,\beta, d)$ be a domain defined by
\begin{equation}
\Sigma(m,\beta, d)=\{z \in \mathbb{C}^N:
\sum_{k=1}^{n}\left(\sum_{j=s_{k-1}}^{s_k-1}|z_j|^{2d_j}\right)^{\beta_k}<1\}, \nn
\end{equation}
where
$d=(d_1, \dots, d_N) \in \mathbb{N}^N, \;
\beta=(\beta_1, \dots, \beta_n) \in \mathbb{N}^n$, and
$N=m_1+\dots+m_n, s_0=1, s_k=s_{k-1}+m_k$ and $m=(m_1, \dots, m_n)$.
If there exists a proper holomorphic mapping between
$\Sigma(p, \alpha, c)$ and $\Sigma(m, \beta, d)$,
then, up to reordering the variables, $p=m, \; \alpha=\beta$ and
$c_j/d_j \in \mathbb{N}$.

All results mentioned above are for the equi-dimensional case, namely,
source and target domains have the same dimensions.
On the other hand, our present research is to study the higher codimensional case.
By codimension, we mean the difference of the source and the target dimensions.
When we study such a case, we find some strange phenomena.
For instance, by \cite{Do},
there exists a proper holomorphic mapping $f : B^{N} \to B^{N+1}$ that
can be extended continuously to the closure $\ol{B^{N}}$ but
can not be extended in a $C^2$ way to any open subset of the boundary of $B^N$.
The determination of proper holomorphic mappings
between pseudoellipsoids for small codimensional case
was studied by P.~Ebenfelt-D.~N.~Son \cite{ES} and they obtained the explicit parametrization
of the mappings.
In the case of generalized pseudoellipsoids,
under some assumptions on dimensions and nondegeneracy condition of
a certain kind of Jacobian matrix,
the author \cite{Ha2} obtained the conditions to exist mappings between those domains and
gave equivalence classes of the mappings.
The purpose of this article is to weaken those assumptions.
The relations between author's previous results and the present results
are discussed in the final section.

We list some notation.
Let $w_j=(w^{1}_{j}, \dots, w^{m_j}_{j}) \in \mathbb{C}^{m_j},
w=(w_1, \dots, w_N) \in \mathbb{C}^{m_1}
\times \dots \times \mathbb{C}^{m_N}$,
and
$p_j=(p^{1}_{j}, \dots, p^{m_j}_{j}) \in (\mathbb{Z}_{\geq 0})^{m_j}, \;
p=(p_1, \dots, p_N) \in (\mathbb{Z}_{\geq 0})^{m_1}
\times \dots \times (\mathbb{Z}_{\geq 0})^{m_N}$,
and $\alpha=(\alpha_1, \dots, \alpha_{N-1}, 1) \in \mathbb{N}^{N}$.
\begin{itemize}
\item The multi-index notations;
       $(w_{j})^{p_j}=(w^{1}_{j})^{p^{1}_{j}} \cdots (w^{m_j}_{j})^{p^{m_j}_{j}}$
       and
      $(w)^p=(w_1)^{p_1} \\ \times \cdots \times (w_N)^{p_N}$.
\item $|||w|||^{2\alpha}=||w_1||^{2\alpha_1}+ \dots +||w_{N-1}||^{2\alpha_{N-1}}
           +||w_N||^2$.
      $|p_j|=p^{1}_{j}+\dots+p^{m_j}_{j}$ and
      $|p|=|p_1|+\dots+|p_N|$.
\item The total degree of $w_j$ and $\ol{w_j}$ in a monomial $(w_j)^{p_j} \times (\ol{w_j})^{q_j}$
          is the sum $|p_j|+|q_j|$.
\item $\pa \Omega$ is the boundary of the domain $\Omega$.
         $\pa^{\ast}\Omega$ is the strongly pseudoconvex part of $\pa \Omega$.
\item $\text{Re}~z=x$.
\end{itemize}
When we need to distinguish the notation of source and
target objects, we add `tilde' on the target objects.

The author would like to express his sincere thanks to Professor Akio Kodama for
reading this article and pointing out some mistakes.
His advice was very  valuable and made some arguments clear.

\section{decomposition of a CR vector bundle}
Following \cite{MM}, we decompose a CR vector bundle.
We employ the usual coordinate
$(x, w) \in \mathbb{R} \times \mathbb{C}^m$
on the boundary of $\Hm$, that is,
we identify naturally $(x, w)$ with $(x+i|||w|||^{2\alpha}, w)$
throughout this paper.

We define CR vector fields $L^{\lambda}_{j}$ by
\begin{equation}
L^{\lambda}_{j}=\dfrac{\pa}{\pa w^{\lambda}_{j}}+i\alpha_j||w_j||^{2(\alpha_j-1)}
   \ol{w^{\lambda}_{j}}\dfrac{\pa}{\pa x}, \qquad
   j=1, \dots, N, \quad \lambda=1, \dots, m_j. \nn
\end{equation}
Define the pseudohermitian structure $\vartheta$ as
\begin{equation}
\vartheta=dx-i\sum^{N}_{j=1} \sum^{m_j}_{\mu=1}
\alpha_j ||w_j||^{2(\alpha_j-1)}
   \left(\ol{w^{\mu}_{j}}dw^{\mu}_{j}-
   w^{\mu}_{j}d\ol{w^{\mu}_{j}} \right). \nn
\end{equation}
We define
the Levi form on $\pa \Hm$ to be a $2$-form:
$\text{Levi}=-id\vartheta$.

We introduce the holomorphic vector fields on $\pa^{\ast}\Hm$:
\begin{equation}
E_j=\dfrac{1}{\alpha_j}\sum^{m_j}_{\mu=1}w^{\mu}_{j}L^{\mu}_{j}, \; j=1, \dots, N,    \nn
\end{equation}
which is called a vector field of radial type.
The vector fields $E_1, \dots, E_N$ span an $N$-dimensional
subbundle $\mathcal{E} \subset T^{1,0} \pa^{\ast} \Hm$.
We denote by $\mathcal{E}^{\bot}$ the orthogonal component of
$\mathcal{E}$ in $T^{1,0}\pa^{\ast} \Hm$ with respect to the Levi form.

Let
$Q:T^{1,0}\pa^{\ast} \Hm \to \mathcal{E}^{\bot}$
be the projection
\begin{equation}
Q(X)=X-\sum^{N}_{j=1}\dfrac{\text{Levi}(X, \ol{E_j})}
{\text{Levi}(E_j, \overline{E_j})}E_j. \nn
\end{equation}
Let $W^{\lambda}_{j}=Q(L^{\lambda}_{j})$.
Then, by calculation,
it is written as
\begin{align}
W^{\lambda}_{j}=&\dfrac{\pa}{\pa w^{\lambda}_{j}}-\sum^{m_j}_{\mu=1}
\dfrac{\ol{w^{\lambda}_{j}}w^{\mu}_{j}}{||w_j||^2}
\dfrac{\pa}{\pa w^{\mu}_{j}} \nn \\
=&\sum^{m_j}_{\mu=1}
\left\{\delta^{\lambda}_{\mu}-\dfrac{\ol{w^{\lambda}_{j}}w^{\mu}_{j}}{||w_j||^2}
\right\}L^{\mu}_{j}.  \label{wexp}
\end{align}
Here $\delta^{\lambda}_{\mu}$ is the Kronecker's delta.
Then the bundle $\mathcal{E}^{\bot}$ is an $(m-N)$-dimensional subbundle of
$T^{1,0}\pa^{\ast} \Hm$ and is generated by the set of vector fields
$\{W^{\lambda}_{j}\}$.

Let $\mathcal{W}_j$ be the subbundle of $T^{1,0}\pa^{\ast} \Hm$ generated by
$W^{1}_{j}, \dots, W^{m_j}_{j}$.
Since the relation $w^{1}_{j}W^{1}_{j}+\dots+w^{m_j}_{j}W^{m_j}_{j}=0$ holds,
the subbundle $\mathcal{W}_j$ is, in fact, generated by, say,
$W^{1}_{j}, \dots, W^{m_j-1}_{j}$.
Then we have
$\mathcal{E}^{\bot}=\mathcal{W}_1 \oplus \dots \oplus \mathcal{W}_N$
and
$\mathcal{W}_j \bot \mathcal{W}_k$ if $j \ne k$.
Hence we obtain the orthogonal decomposition
\begin{equation}
T^{1, 0}\pa^{\ast} \Hm=\mathcal{W}_1 \oplus \dots \oplus \mathcal{W}_N \oplus \mathcal{E}.
\label{ds}
\end{equation}
We sometimes use the notation $\mathcal{E}=\mathcal{W}_{N+1}$.
For later use, we here list the values of the Levi form as follows:
Using the fact that
$\theta(L^{\lambda}_{j})=0$ and
\begin{equation}
\theta([L^{\lambda}_{j}, \ol{L^{\mu}_{j}}])=-2i\alpha_j||w_j||^{2(\alpha_j-1)}
 \Big\{\delta^{\mu}_{\lambda}+(\alpha_j-1)\dfrac{\ol{w^{\lambda}_{j}}w^{\mu}_{j}}{||w_j||^{2}}
\Bigr\}, \nn
\end{equation}
we obtain that
\begin{equation}
\begin{split}
&\text{Levi}(W^{\lambda}_{j}, \ol{W^{\mu}_{j}})=-2\alpha_j||w_j||^{2(\alpha_j-1)}
     \dfrac{\ol{w^{\lambda}_{j}}w^{\mu}_{j}}{||w_j||^{2}},  \quad
\text{Levi}(W^{\lambda}_{j}, \ol{W^{\mu}_{k}})=0, \; j \neq k,  \\
&\text{Levi}(W^{\lambda}_{j}, \ol{E_k})=0, \quad
\text{Levi}(E_j, \ol{E_j})=2||w_j||^{2\alpha_j}.  \label{levical}
\end{split}
\end{equation}
Rest of this section is devoted to obtain relations among
$\W_j$'s, $\tldW_i$'s and $\Phi_{\ast}$.
In the following, for $j, k=1,\dots, N$, we denote by $\wt{W}^{\lambda}_{j, k}$
(resp.$\wt{W}^{\lambda}_{j, N+1}$)
the $\tldW_k$-component (resp. the $\wt{\mathcal{E}}$-component)
of $\Phi_{\ast}W^{\lambda}_{j}$.
For a fixed $j=1, \dots, N$, also denote by $\wt{W}_{N+1, k},$ $k=1, \dots, N$
(resp. $\wt{W}_{N+1, N+1}$)
the $\tldW_k$-component (resp. the $\wt{\mathcal{E}}$-component)
of \\ $\Phi_{\ast}E_j$.
Let $J_i$ be a subset of $\{1, \dots, N\}$ defined by
$J_i=\{\ell \in \{1, \dots, N\} : \tldW_{\ell}\text{-} \\ \text{component of }
\Phi_{\ast}\mathcal{W}_i \text{ is equal to zero} \}$.
\begin{lemma}\label{indexdecomposition}
For any $j, k=1, \dots, N$ with $j \neq k$,  we have
$J_j \cap J_k=\emptyset$.
\end{lemma}
\begin{proof}
Take any $W^{\lambda}_{j} \in \W_j$ and $W^{\mu}_{k} \in \W_k$
for $j, k=1, \dots, N$ with $j \neq k$.
Let us denote by
\begin{align}
&\Phi_{\ast}W^{\lambda}_{j}=\wt{W}_{j,1}^{\lambda}+ \dots+
        \wt{W}_{j, N}^{\lambda}+\wt{W}_{j, N+1}^{\lambda},  \label{phiwj}\\
&\Phi_{\ast}W^{\mu}_{k}=\wt{W}_{k, 1}^{\mu}+ \dots+
        \wt{W}_{k, N}^{\mu}+\wt{W}_{k, N+1}^{\mu}. \label{phiwk}
\end{align}
We calculate the Levi form
$\widetilde{\text{Levi}}(\Phi_{\ast}W^{\lambda}_{j}, \ol{\Phi_{\ast}W^{\mu}_{k}})$
in two ways.
Since the mapping $\Phi$ is CR, there exists a positive function $f$ such that
$\Phi^{\ast}\widetilde{\vartheta}=f\vartheta$.
Hence we have
\begin{equation}
\widetilde{\text{Levi}}(\Phi_{\ast}W^{\lambda}_{j}, \ol{\Phi_{\ast}W^{\mu}_{k}})
=f\text{Levi}(W^{\lambda}_{j}, \ol{W^{\mu}_{k}})=0.  \nn
\end{equation}
On the other hand, by using (\ref{phiwj}) and (\ref{phiwk}),
it is expanded as follows:
\begin{align}
&\widetilde{\text{Levi}}(\Phi_{\ast}W^{\lambda}_{j}, \ol{\Phi_{\ast}W^{\mu}_{k}})
    \label{leviexp} \\
&=\widetilde{\text{Levi}}(\wt{W}^{\lambda}_{j, 1}, \ol{\wt{W}^{\mu}_{k, 1}})+ \dots+
    \widetilde{\text{Levi}}(\wt{W}^{\lambda}_{j, N}, \ol{\wt{W}^{\mu}_{k, N}})+
   \widetilde{\text{Levi}}(\wt{W}^{\lambda}_{j, N+1}, \ol{\wt{W}^{\mu}_{k, N+1}})
     \nn \\
   & \quad
   +\wt{\text{Levi}}(\wt{W}^{\lambda}_{j, 1}, \ol{\wt{W}^{\mu}_{k, N+1}})+\dots
  +\wt{\text{Levi}}(\wt{W}^{\lambda}_{j, N}, \ol{\wt{W}^{\mu}_{k, N+1}}) \nn  \\
& \quad
    +\wt{\text{Levi}}(\wt{W}^{\lambda}_{j, N+1}, \ol{\wt{W}^{\mu}_{k, 1}})+\dots
    +\widetilde{\text{Levi}}(\wt{W}^{\lambda}_{j, N+1}, \ol{\wt{W}^{\mu}_{k, N}}).   \nn
\end{align}
For $i=1, \dots, N$, let us denote by
\begin{align}
&\wt{W}^{\lambda}_{j, i}=a^{1}_{i}\wt{W}^{1}_{i}+\dots+
     a^{n_{i}-1}_{i}\wt{W}^{n_{i}-1}_{i},  \quad
\wt{W}^{\mu}_{k, i}=b^{1}_{i}\wt{W}^{1}_{i}+\dots+
     b^{n_{i}-1}_{i}\wt{W}^{n_{i}-1}_{i}, \nn  \\
&\wt{W}^{\lambda}_{j, N+1}=A_1\wt{E}_1+\dots+A_N\wt{E}_N,  \quad
\wt{W}^{\mu}_{k, N+1}=B_1\wt{E}_1+\dots+B_N\wt{E}_N,  \nn
\end{align}
then it follows from (\ref{levical}) that each term on the right hand side of (\ref{leviexp})
is calculated as follows:
\begin{align}
&\wt{\text{Levi}}(\wt{W}^{\lambda}_{j, i}, \ol{\wt{W}^{\mu}_{k, i}})
=-2\beta_{i}||\wt{w}_{i}||^{2(\beta_{i}-1)}
\sum_{\nu_1, \nu_2=1}^{n_{i}-1}a^{\nu_1}_{i}\ol{b^{\nu_2}_{i}}
\dfrac{\ol{\wt{w}^{\nu_1}_{i}} \wt{w}^{\nu_2}_{i}}
      {||\widetilde{w}_{i}||^2}, \; i=1, \dots, N,   \nn  \\
&\wt{\text{Levi}}(\wt{W}^{\lambda}_{j, N+1}, \ol{\wt{W}^{\mu}_{k, N+1}})
=2(A_1\ol{B_1}||\wt{w}_1||^{2\beta_1}+\dots+A_N\ol{B_N}||\wt{w}_N||^{2}),   \nn \\
&\wt{\text{Levi}}(\wt{W}^{\lambda}_{j, i}, \ol{\wt{W}^{\mu}_{k, N+1}})=
\widetilde{\text{Levi}}(\wt{W}^{\lambda}_{j, N+1}, \ol{\wt{W}^{\mu}_{k, i}})=0, \; i=1, \dots, N.  \nn
\end{align}
Hence we obtain
\begin{equation}
-\sum^{N}_{i=1}2\beta_{i}||\wt{w}_{i}||^{2(\beta_{i}-1)}
\sum_{\nu_1, \nu_2=1}^{n_{i}-1}a^{\nu_1}_{i}\ol{b^{\nu_2}_{i}}
\dfrac{\ol{\wt{w}^{\nu_1}_{i}} \wt{w}^{\nu_2}_{i}}
      {||\widetilde{w}_{i}||^2}
+2\sum^{N}_{i=1}A_{i}\ol{B_i}||\wt{w}_i||^{2\beta_i}=0, \nn
\end{equation}
which means that for each $i=1, \dots, N$,
\begin{align}
&a^{1}_{i}\ol{b^{1}_{i}}=\dots=a^{1}_{i}\ol{b^{n_{i}-1}_{i}}=0, \nn\\
& \qquad \vdots \nn \\
&a^{n_{i}-1}_{i}\ol{b^{1}_{i}}=\dots=a^{n_{i}-1}_{i}\ol{b^{n_{i}-1}_{i}}
=0.  \nn
\end{align}
Take any $i=1, \dots, N$ and fix it.
If $a^{\lambda}_{i}=0$ for any $\lambda=1, \dots, n_{i}-1$, then
$\wt{W}^{\lambda}_{j, i}=0$; and hence, the $\tldW_i$-component of $\Phi_{\ast}W^{\lambda}_{j}$
is equal to zero.
If there exists $\lambda=1, \dots, n_{i}-1$ such that
$a^{\lambda}_{i}\neq0$, then
$b^{1}_{i}=\dots=b^{n_{i}-1}_{i}=0$ and
this means that
the $\tldW_i$-component of
$\Phi_{\ast}W^{\mu}_{k}$ is equal to zero.
Hence both $\Phi_{\ast}W^{\lambda}_{j}$ and $\Phi_{\ast}W^{\mu}_{k}$ do not have
the $\tldW_i$-component simultaneously.
\end{proof}
The same conclusion holds for
$\Phi_{\ast}\mathcal{E}$ and $\Phi_{\ast}\mathcal{W}_j$.
\begin{lemma}
For any $j=1, \dots, N$, we have $J_j \cap J_{N+1}=\emptyset$.
\end{lemma}
\begin{proof}
Take any $k=1, \dots, N$ and fix it.
Let us denote by
\begin{align}
&\Phi_{\ast}E_k=\wt{W}_{N+1, 1}+\dots+\wt{W}_{N+1, N}+\wt{W}_{N+1, N+1}, \nn \\
&\Phi_{\ast}W^{\lambda}_{j}=\wt{W}^{\lambda}_{j, 1}+\dots+\wt{W}^{\lambda}_{j, N}
+\wt{W}^{\lambda}_{j, N+1}.  \nn
\end{align}
Lemma follows by the same argument as in Lemma~{\ref{indexdecomposition}}.
\end{proof}

\begin{lemma}\label{nowe}
There does not exist
$W^{\lambda}_{j} \in \mathcal{W}_j, \;  j=1, \dots, N $ such that
$\Phi_{\ast}W^{\lambda}_{j} \in \wt{\mathcal{E}}$.
\end{lemma}

\begin{proof}
We use the expansion of $G^{\mu}_{\ell}$:
\begin{equation}
G^{\mu}_{\ell}(x, w)=\sum_{|p|+q \geq 0}b^{\mu}_{\ell, p, q}(w)^p
   (x+i|||w|||^{2\alpha})^q.  \nn
\end{equation}
Suppose that there exists $W^{\lambda}_{j} \in \mathcal{W}_j$
such that $\Phi_{\ast}W^{\lambda}_{j} \in \wt{\mathcal{E}}$
and express it as
$\Phi_{\ast}W^{\lambda}_{j}=A^{\lambda}_1\wt{E}_1+\dots+A^{\lambda}_N\wt{E}_N$.
Take $\ell=1, \dots, N$ such that $A^{\lambda}_{\ell} \neq 0$.
Apply $\wt{w}^{\mu}_{\ell}=G^{\mu}_{\ell}$ to this relation to get the following:
\begin{align}
\dfrac{A^{\lambda}_{\ell}}{\beta_{\ell}}G^{\mu}_{\ell}=
W^{\lambda}_{j}G^{\mu}_{\ell}
&=\sum^{m_j}_{\nu=1}\left(\delta^{\nu}_{\lambda}-\dfrac{\ol{w^{\lambda}_{j}}w^{\nu}_{j}}{||w_j||^2}\right)
   L^{\nu}_{j}G^{\mu}_{\ell}   \label{wg}  \\
&=\sum_{\substack{|p|+q \geq 0 \\ p^{\lambda}_{j}\geq 1}}
b^{\mu}_{\ell, p, q}\dfrac{\pa (w)^{p}}{\pa w^{\lambda}_{j}}(x+i|||w|||^{2\alpha})^q  \nn \\
& \qquad -\dfrac{\ol{w^{\lambda}_{j}}}{||w_j||^2}
\sum^{m_j}_{\nu=1}\sum_{\substack{|p|+q \geq 0 \\ p^{\nu}_j\geq 1}}
b^{\mu}_{\ell, p, q}p^{\nu}_j(w)^p
(x+i|||w|||^{2\alpha})^q. \nn
\end{align}
For convenience, we want to rewrite the last summation in (\ref{wg}) as follows.
To this end, let us introduce a subset $P_k$ of $\mathbb{Z}^{m_{j}}$ defined by
\begin{align}
P_k=\{(p^{1}_{j}, \dots, p^{m_j}_j) \in \mathbb{Z}^{m_j} : \;
&k \; \text{components among} \; p^{1}_{j}, \dots, p^{m_j}_{j} \;
\text{are positive and }  \nn \\
&\text{the rest are zero} \}.   \nn
\end{align}
Take any $p^0_j \in P_{k}$ such that
its $\lambda_1$-th, $\dots, \lambda_{k}$-th
components are positive and the others are zero.
Then, among $m_j$ summations in the second term on the far right hand side of (\ref{wg}),
\begin{equation}
\sum_{\substack{|p|+q \geq 0 \\ p^{1}_j\geq 1}}
b^{\mu}_{\ell, p, q}p^{1}_j(w)^p
(x+i|||w|||^{2\alpha})^q, \dots,
\sum_{\substack{|p|+q \geq 0 \\ p^{m_j}_j\geq 1}}
b^{\mu}_{\ell, p, q}p^{m_j}_j(w)^p
(x+i|||w|||^{2\alpha})^q, \nn
\end{equation}
the term of the form:
\begin{equation}
\sum_{\substack{|p|+q \geq 0 \\ p_j=p^{0}_{j}}}  \nn
b^{\mu}_{\ell, p, q}p^{\nu}_j(w)^p
(x+i|||w|||^{2\alpha})^q
\end{equation}
comes only from
the $\lambda_1$-th, $\dots$, the $\lambda_k$-th summations.
Thus (\ref{wg}) can be rewritten as
\begin{align}
\dfrac{A^{\lambda}_{\ell}}{\beta_{\ell}}G^{\mu}_{\ell}=
&\sum_{\substack{|p|+q \geq 0 \\ p^{\lambda}_{j}\geq 1}}
b^{\mu}_{\ell, p, q}\dfrac{\pa (w)^{p}}{\pa w^{\lambda}_{j}}(x+i|||w|||^{2\alpha})^q  \nn \\
&\qquad -\dfrac{\ol{w^{\lambda}_{j}}}{||w_j||^2}
\Bigl\{m_j\sum_{\substack{|p|+q \geq 0 \\ p_j \in P_{m_j}}}
b^{\mu}_{\ell, p, q}|p_j|(w)^p (x+i|||w|||^{2\alpha})^q \nn \\
& \qquad \qquad +\dots+
(m_j-k)
\sum_{\substack{|p|+q\geq 0 \\ p_j \in P_{m_j-k}}}
b^{\mu}_{\ell, p, q}|p_j|(w)^p(x+i|||w|||^{2\alpha})^q \nn \\
&\qquad \qquad +\dots+
\sum_{\substack{|p|+q \geq 0 \\ p_j \in P_1}}
b^{\mu}_{\ell, p, q}|p_j|(w)^p(x+i|||w|||^{2\alpha})^q \Bigr\}.  \nn
\end{align}
Hence comparing the $\ol{w^{\lambda}_{j}}$ terms and the $(w)^p$ terms with
$p_j \in P_{m_j-k}$,
we have
\begin{equation}
\sum_{\substack{|p|+q \geq 0 \\ p_j \in P_{m_j-k}}}b^{\mu}_{\ell, p, q}|p_j|(w)^p
(x+i|||w|||^{2\alpha})^q=0, \quad  k=0, \dots, m_j-1 \nn
\end{equation}
and thus $G^{\mu}_{\ell}$ has an expansion
\begin{equation}
G^{\mu}_{\ell}(x, w)=\sum_{\substack{|p|+q \geq 0 \\ p_j=0}}b^{\mu}_{\ell, p, q}(w)^p
   (x+i|||w|||^{2\alpha})^q.  \nn
\end{equation}
This means that the variables $w_j$ do not appear in $(w)^p$.
Therefore again by the relation (\ref{wg}) and by $A^{\lambda}_{\ell} \neq 0$,
we conclude that $G^{\mu}_{\ell}=0$
for any $\mu=1, \dots n_{\ell}$. This contradicts the assumption on the main theorem.
\end{proof}

From the proof of this lemma, the following corollary holds, which will be used
in the next argument.
\begin{corollary}\label{f0}
Let $f$ be a CR function on $\pa \Hm$.
Assume that there exist $W^{\lambda}_{j} \in \mathcal{W}_j$ and a constant $A$
such that $W^{\lambda}_{j}f=Af$.
If $A\neq 0$, then $f\equiv0$ and if $A=0$, then
$f$ has an expansion:
\begin{equation}
f=\sum_{\substack{|p|+q \geq 0 \\ p_j=0}} a_{p, q}(w)^p(x+i|||w|||^{2\alpha})^q. \nn
\end{equation}
\end{corollary}

From Lemma~{\ref{indexdecomposition}} to Lemma~{\ref{nowe}},
we have proved that there exists a permutation $\sigma$ of $\{1, \dots, N\}$
such that
$\Phi_{\ast}\mathcal{W}_{\sigma(j)} \subset \tldW_j \oplus \wt{\mathcal{E}}$.
Here we assert that
$\Phi_{\ast}\mathcal{W}_{\sigma(j)} \subset \tldW_j$ for all $j=1, \dots, N$.
We now verify this only in the case where $j=1$, since the verification in the general case
is almost identical.
So, take any $W^{\lambda}_{\sigma(1)} \in \mathcal{W}_{\sigma(1)}$ and write
\begin{equation}
\Phi_{\ast}W^{\lambda}_{\sigma(1)}=a^{\lambda}_1\wt{W}^{1}_{1}+\dots+a^{\lambda}_{n_1-1}\wt{W}^{n_1-1}_{1}
+A^{\lambda}_1\wt{E}_1+\dots+A^{\lambda}_N\wt{E}_N.  \label{wwe}
\end{equation}
Assume that there exist $\lambda=1, \dots, m_{\sigma(1)}$ and $\ell=2, \dots, N$
such that $A^{\lambda}_{\ell} \neq 0$.
Apply $\wt{w}^{\mu}_{\ell}=G^{\mu}_{\ell}$ to (\ref{wwe}) to get
$W^{\lambda}_{\sigma(1)}G^{\mu}_{\ell}=(A^{\lambda}_{\ell}/\beta_{\ell})G^{\mu}_{\ell}$.
Then by Corollary~{\ref{f0}}, $G^{\mu}_{\ell}=0$ for any $\mu=1, \dots, n_{\ell}$,
which is a contradiction.
Thus $A^{\lambda}_{\ell}=0$ for any  $\lambda=1, \dots, m_{\sigma(1)}$ and $\ell=2, \dots, N$.
Among $A^{1}_{1}, \dots, A^{m_{\sigma(1)}}_{1}$,
assume that $A^{\lambda_1}_{1}, \dots, A^{\lambda_i}_{1} \neq 0$
and the others are zero.
Apply $\wt{x}=(F+\ol{F})/2$ to (\ref{wwe}) to get
$(1/2)W^{\lambda}_{\sigma(1)}F=iA^{\lambda}_{1}||G_1||^{2\beta_1}$.
Since $F$ has an expansion
\begin{equation}
F=\sum_{|p|+q \geq 0}a_{p, q}(w)^p(x+i|||w|||^{2\alpha})^q,   \label{fwx}
\end{equation}
we get
\begin{align}
&\dfrac{1}{2}\sum_{\substack{|p|+q \geq 0 \\ p^{\lambda}_{\sigma(1)} \geq 1}}a_{p, q}
\dfrac{\pa (w)^p}{\pa w^{\lambda}_{\sigma(1)}}(x+i|||w|||^{2\alpha})^q  \label{anylambda} \\
&-\dfrac{1}{2}\dfrac{\ol{w^{\lambda}_{\sigma(1)}}}{||w_{\sigma(1)}||^2}
\sum^{m_{\sigma(1)}}_{\nu=1}
\sum_{\substack{|p|+q \geq 0 \\ p^{\nu}_{\sigma(1)}\geq 1}}a_{p, q}p^{\nu}_{\sigma(1)}
(w)^p(x+i|||w|||^{2\alpha})^q  \nn \\
&=iA^{\lambda}_{1}||G_1||^{2\beta_1}  \nn
\end{align}
for any $\lambda=1, \dots, m_{\sigma(1)}$.
Multiply (\ref{anylambda}) by $w^{\lambda}_{\sigma(1)}$ and add them from
$\lambda=1$ to $\lambda=m_{\sigma(1)}$.
Then we have
\begin{equation}
0=i||G_1||^{2\beta_1}\{A^{\lambda_1}_{1}w^{\lambda_1}_{\sigma(1)}+\dots+
             A^{\lambda_i}_{1}w^{\lambda_i}_{\sigma(1)}\}  \nn
\end{equation}
and this means that $G_1=0$. This is a contradiction.
Thus $A^{\lambda}_{1}=0$ for all $\lambda$.
Hence, the $\wt{\mathcal{E}}$-component of $\Phi_{\ast}W^{\lambda}_{\sigma(1)}$
is equal to zero and
$\Phi_{\ast}\mathcal{W}_{\sigma(1)} \subset \tldW_1$, as asserted.
By the same argument as above, it can be seen that
$\Phi_{\ast}\mathcal{E} \subset \wt{\mathcal{E}}$.
Summarizing the above, we obtain the following:
\begin{proposition}\label{sigmae}
There exists a permutation $\sigma$ of $\{1, \dots, N\}$ such that
$\Phi_{\ast}\mathcal{W}_{\sigma(j)} \subset \tldW_j$ for all $j=1, \dots, N$; and
$\Phi_{\ast}\mathcal{E} \subset \wt{\mathcal{E}}$.
\end{proposition}

\section{Expansions of $F$ and $G$}
Using Proposition~{\ref{sigmae}}, we can characterize the expansions of
$F$ and $G^{\lambda}_{j}$.

\begin{lemma}\label{fgex}
Up to an automorphism of $\Hn$,
the mapping $(F, G)$ has the following expansions  on $\pa \Hm$:
\begin{align}
&F(x, w)=x+i|||w|||^{2\alpha},    \label{fx} \\
&G^{\mu}_{j}(w)
=\sum_{|p_{\sigma(j)}| \geq 1}b^{\mu}_{j, p_{\sigma(j)}}(w_{\sigma(j)})^{p_{\sigma(j)}}, \; j=1, \dots, N.
\label{gw}
\end{align}
\end{lemma}
\begin{proof}
\begin{claim}\label{claimg}
$G^{\mu}_{j}$ is expanded as
\begin{equation}
G^{\mu}_{j}(w)
=\sum_{|p_{\sigma(j)}| \geq 0}b^{\mu}_{j, p_{\sigma(j)}}(w_{\sigma(j)})^{p_{\sigma(j)}}.
\label{gex1st}
\end{equation}
\end{claim}
Take any $j=1, \dots, N$ and fix it.
Since $\Phi_{\ast}W^{\lambda}_{\sigma(i)} \in \tldW_i$,
by applying $\wt{w}^{\mu}_{j}=G^{\mu}_{j}$ to
$\Phi_{\ast}W^{\lambda}_{\sigma(i)}, i=1, \dots, N, \lambda=1, \dots, m_{\sigma(i)}$ with $i \neq j$,
we get the relations:
\begin{align}
&W^{1}_{\sigma(1)}G^{\mu}_{j}=0, \dots, W^{m_{\sigma(1)}}_{\sigma(1)}G^{\mu}_{j}=0,
\nn \\
& \qquad \vdots   \nn \\
&W^{1}_{\sigma(N)}G^{\mu}_{j}=0, \dots, W^{m_{\sigma(N)}}_{\sigma(N)}G^{\mu}_{j}=0.
\nn
\end{align}
Note that
$W^{1}_{\sigma(j)}G^{\mu}_{j}=0, \dots, W^{m_{\sigma(j)}}_{\sigma(j)}G^{\mu}_{j}=0$
are not contained in these relations.
It follows from Corollary~{\ref{f0}} that we obtain the expansion:
\begin{equation}
G^{\mu}_{j}(x, w)
=\sum_{|p_{\sigma(j)}|+q \geq 0}b^{\mu}_{j, p_{\sigma(j)}, q}
    (w_{\sigma(j)})^{p_{\sigma(j)}}(x+i|||w|||^{2\alpha})^q.  \nn
\end{equation}
We apply $\wt{w}^{\mu}_{j}=G^{\mu}_{j}$ to
$\Phi_{\ast}E_k=A^k_1\wt{E}_1+\dots+A^k_N\wt{E}_N$
with $k \neq \sigma(j)$.
Assume that $A^{k}_{j} \neq 0$ for all $k=1, \dots, N$ and $k \neq \sigma(j)$.
Then we get the relation
$E_kG^{\mu}_{j}=A^k_j\wt{E}_j\wt{w}^{\mu}_{j}=(A^k_j/\beta_j)G^{\mu}_{j}$.
This relation is rewritten as
\begin{equation}
2i||w_k||^{2\alpha_k}\sum_{\substack{|p_{\sigma(j)}|+q \geq 0 \\ q \geq 1}}b^{\mu}_{j, p_{\sigma(j), q}}
(w_{\sigma(j)})^{p_{\sigma(j)}}q(x+i|||w|||^{2\alpha})^{q-1}
=\dfrac{A^k_j}{\beta_j}G^{\mu}_{j}.    \nn
\end{equation}
By comparing the terms with $||w_k||^{2\alpha_k}$,
the both sides must be zero.
Since $G^{\mu}_{j}$ does not vanish identically, $A^k_j=0$.
This is a contradiction.
Thus there exists $k=1, \dots, N$ such that $k \neq \sigma(j)$
and $A^{k}_{j}=0$.
Take such $k$ and apply $\wt{w}^{\mu}_{j}=G^{\mu}_{j}$ to
$\Phi_{\ast}E_k=A^k_1\wt{E}_1+\dots+A^k_N\wt{E}_N$
to get the relation $E_kG^{\mu}_{j}=0$.
This is rewritten as
\begin{equation}
2i||w_k||^{2\alpha_k}\sum_{\substack{|p_{\sigma(j)}|+q \geq 0 \\ q \geq 1}}b^{\mu}_{j, p_{\sigma(j), q}}
(w_{\sigma(j)})^{p_{\sigma(j)}}q(x+i|||w|||^{2\alpha})^{q-1}=0.  \nn
\end{equation}
This means that the coefficients $b^{\mu}_{j, p_{\sigma(j)}, q}$ vanish for
$|p_{\sigma(j)}|+q \geq 0, q \geq 1$.
Now we conclude that
$G^{\mu}_{j}$ has an expansion of the form (\ref{gex1st}).

\vspace{1\baselineskip}

Let us denote by
$\Phi_{\ast}E_{\sigma(1)}=A^{\sigma(1)}_{1}\wt{E}_1+\dots+A^{\sigma(1)}_{N}\wt{E}_N$.
Apply $\wt{w}^{\mu}_{j}=G^{\mu}_{j}$ with $j \neq 1$ to get
$(A^{\sigma(1)}_{j}/\beta_j)G^{\mu}_{j}=0$ and this means that
$A^{\sigma(1)}_{j}=0$ for $j \neq 1$.
By the same reason, we have $A^{\sigma(k)}_{j}=0$ for $j\neq k$, namely, we have
$\Phi_{\ast}E_{\sigma(j)}=A^{\sigma(j)}_j\wt{E}_j$, which will be used in the next claim.
\begin{claim}
$F$ is expanded as
\begin{equation}
F(x, w)=a_0+a_1(x+i|||w|||^{2\alpha}). \label{fex1st}
\end{equation}
\end{claim}
First of all we apply $\wt{x}=(F+\ol{F})/2$ to
$\Phi_{\ast}W^{\lambda}_{\sigma(1)} \in \tldW_1, \dots, \Phi_{\ast}W^{\lambda}_{\sigma(N)} \in \tldW_N$
to get
\begin{align}
&W^{1}_{\sigma(1)}F=0, \dots, W^{m_{\sigma(1)}}_{\sigma(1)}F=0,
\nn \\
& \qquad \vdots   \nn \\
&W^{1}_{\sigma(N)}F=0, \dots, W^{m_{\sigma(N)}}_{\sigma(N)}F=0.
\nn
\end{align}
By the same argument as in Claim~{\ref{claimg}}, $F$ is a function of variable $x+i|||w|||^{2\alpha}$.
Next applying  $\wt{x}=(F+\ol{F})/2$ to
$\Phi_{\ast}E_{\sigma(j)}=A^{\sigma(j)}_j\wt{E}_j$
and using the expansion (\ref{fwx}), we get
\begin{equation}
||w_{\sigma(j)}||^{2\alpha_{\sigma(j)}}\sum_{q \geq 1}a_{p, q}q(x+i|||w|||^{2\alpha})^{q-1}
=A^{\sigma(j)}_j||G_j||^{2\beta_j}. \label{fex2}
\end{equation}
Since the right hand side does not contain the variable $x$, we obtain $a_{p, q}=0$ for $q \geq 2$.
This leads to an expansion of the form (\ref{fex1st}).

\vspace{1\baselineskip}

Go back to the proof of lemma.
The equation (\ref{fex2}) is reduced to
$||w_{\sigma(j)}||^{2\alpha_{\sigma(j)}}a_1 \\ =A^{\sigma(j)}_j||G_j||^{2\beta_j}$.
This means that $G^{\mu}_{j}$ does not have a constant term,
which implies the desired expansion of (\ref{gw}).
On the boundary $\pa \Hm$, we have
\begin{equation}
\text{Im} \; (a_0+a_1(x+i|||w|||^{2\alpha}))=|||G|||^{2\beta}.
\end{equation}
Since the right hand side does not have the constant term and the terms containing $x$,
we obtain that $a_0$ and $a_1$ are real and also obtain that $a_1 >0$.
Since the mapping
\begin{equation}
(\wt{z}, \wt{w}) \mapsto (\dfrac{\wt{z}-a_0}{a_1},
\dfrac{\wt{w}_1}{a^{1/2\beta_{\sigma(1)}}_1}, \dots,
\dfrac{\wt{w}_{N-1}}{a^{1/2\beta_{\sigma(N-1)}}_1}, \dfrac{\wt{w}_N}{a^{1/2}_1})
\end{equation}
is an automorphism of $\Hn$.
It follows that $F$ has an desired expansion (\ref{fx}).
This automorphism does not change the form of (\ref{gw}).
\end{proof}

\section{The relations between exponents}
Making use of the expansions of $F$ and $G^{\lambda}_{j}$,
we will get the relations between exponents $\alpha_j$ and $\beta_j$.
It follows from Lemma~{\ref{fgex}} that we have $|||w|||^{2\alpha}=|||G|||^{2\beta}$
on the boundary of $\Hm$.
Take $j$ with $\sigma(j)=N$.
We restrict $|||w|||^{2\alpha}=|||G|||^{2\beta}$ to
the variety
$w_1=\dots=w_{N-1}=0$.
Then it is reduced to
$||w_N||^{2}=||G_j(w_{N})||^{2\beta_j}$.
Substitute the expansion of $G^{\lambda}_j(w_{N})$ into this and
compare the degree of $w_N$.
Then we obtain $\beta_j=1$, which implies that $j=N$.
Hence we get $\sigma(N)=N$.
Thus, from now on, we consider $\sigma$ as a permutation of $\{1, \dots, N-1\}$.

Now we obtain the relations between the exponents $\alpha$ and $\beta$.
\begin{lemma}\label{alphabeta}
There exist $M_1, \dots, M_{N-1} \in \mathbb{N}$ such that
$\alpha_{\sigma(j)}=M_{j}\beta_j$ for $j=1, \dots, N-1$.
\end{lemma}

\begin{proof}
Once we get the relation between $\alpha_{\sigma(1)}$ and $\beta_1$, the relations between
$\alpha_{\sigma(j)}$ and $\beta_j$ for $2 \leq j \leq N-1$
are obtained by the same argument.
So, it suffices to show the case $j=1$.
First we restrict the expansions (\ref{fx}) and (\ref{gw}) to the variety
$w_{\sigma(2)}=\dots=w_{\sigma(N-1)}=w_N=0$
and obtain the relation on the boundary $\pa \Hm$:
\begin{equation}
||w_{\sigma(1)}||^{2\alpha_{\sigma(1)}}=
||\sum_{|p_{\sigma(1)}| \geq 1}b_{1, p_{\sigma(1)}}
(w_{\sigma(1)})^{p_{\sigma(1)}}||^{2\beta_1}. \label{ab}
\end{equation}

Case 1: $\alpha_{\sigma(1)}<\beta_1$.
Comparing the minimal total degree of
$w_{\sigma(1)}$ and $\ol{w_{\sigma(1)}}$ in the both sides,
we reach a contradiction.

Case 2: $\alpha_{\sigma(1)} \geq \beta_1$.
Assume that there does not exist $M_1 \in \mathbb{N}$ as in the lemma.
Take $M \in \mathbb{N}$  such that
$M\beta_1 < \alpha_{\sigma(1)} <(M+1)\beta_1$.
The terms of $w_{\sigma(1)}$ and $\ol{w_{\sigma(1)}}$ which have total degree
less than or equal to $2M\beta_1$ on the right hand side are zero.
Hence the equation (\ref{ab}) becomes
\begin{equation}
||w_{\sigma(1)}||^{2\alpha_{\sigma(1)}}
=||\sum_{|p_{\sigma(1)}| \geq M+1}
b_{1; p_{\sigma(1)}}
    (w_{\sigma(1)})^{p_{\sigma(1)}}||^{2\beta_1}.
         \nn
\end{equation}
Comparing the minimal total degree of $w_{\sigma(1)}$ and $\ol{w_{\sigma(1)}}$
in the both sides, we reach a contradiction.
Hence we obtain $\alpha_{\sigma(1)}=M_1\beta_1$ for some $M_1 \in \mathbb{N}$.
This completes the proof.
\end{proof}

\section{Normalization of the mapping}
Substituting $\alpha_{\sigma(1)}=M_1 \beta_1$ to (\ref{ab}) and
comparing the total degree of
$w_{\sigma(1)}$ and $\ol{w_{\sigma(1)}}$,
we conclude that $G^{\mu}_{1}$ is of the form:
\begin{equation}
G^{\mu}_{1}(w_{\sigma(1)})=\sum_{|p_{\sigma(1)}|=M_1}
   b^{\mu}_{1;p_{\sigma(1)}}(w_{\sigma(1)})^{p_{\sigma(1)}}.  \nn
\end{equation}
Analogously, we conclude that $G^{\mu}_{j}, j=1, \dots, N,$ have the following expansions:
\begin{align}
&G^{\mu}_{j}(w_{\sigma(j)})=\sum_{|p_{\sigma(j)}|=M_j}
 b^{\mu}_{j, p_{\sigma(j)}}(w_{\sigma(j)})^{p_{\sigma(j)}},
\; j=1, \dots, N-1, \; \mu=1, \dots, n_j, \label{gm} \\
&G^{\mu}_{N}(w_N)=\sum_{|p_N|=1}b^{\mu}_{N, p_N}(w_N)^{p_N},
    \;  \mu=1, \dots, n_N. \nn
\end{align}

We give a normalization of one block case by
restricting $\Phi=(F, G)$ to the variety $w_{\sigma(2)}=\dots=w_{\sigma(N-1)}=w_N=0$.
This restriction does not influence to $G_1$.
The reduced mapping is now
\begin{equation}
(F, G_1):\{\text{Im}z=||w_{\sigma(1)}||^{2\alpha_{\sigma(1)}}\} \to
\{\text{Im}\wt{z}=||\wt{w}_1||^{2\beta_1}\}.
\end{equation}

\begin{lemma}\label{dangelonormal}
Let $\alpha_{\sigma(1)}=M_1\beta_1$. Then we have the following:

$\mathrm{(a)}$ \;
$n_1 \geq \dfrac{(M_1+m_{\sigma(1)}-1)!}{M_1!(m_{\sigma(1)}-1)!}; $ and

$\mathrm{(b)}$ \;
$(F, G_1)$ is equivalent to
$(z, H_{M_1}(w_{\sigma(1)}), 0)$.
\end{lemma}
\begin{proof}
We introduce two notations:
\begin{equation}
T_{M, m}=\dfrac{(M+m-1)!}{M!(m-1)!}=
\begin{pmatrix}
M+m-1 \\
m-1
\end{pmatrix}, \quad
C_{M, p}=\sqrt{\dfrac{M!}{p^{1}! \dots p^{m}!}} \nn
\end{equation}
for $M, m \in \mathbb{N}$ and $p=(p^{1}, \dots, p^{m}) \in \mathbb{N}^m$.
Note that $T_{M, m}$ is the number of solutions of $x_1+\dots+x_{m}=M$
for $x_1, \dots, x_{m} \in \mathbb{Z}_{\geq 0}$.
Substituting (\ref{gm}) and $a_{\sigma(1)}=M_1\beta_1$ into (\ref{ab}) and
taking the $\beta_1$-th root, we get the equation:
\begin{equation}
||w_{\sigma(1)}||^{2M_1}=||\sum_{|p_{\sigma(1)}|=M_1}b_{1;p_{\sigma(1)}}
(w_{\sigma(1)})^{p_{\sigma(1)}}||^2. \label{M1}
\end{equation}
Since the coefficients of
$|(w_{\sigma(j)})^{p_{\sigma(j)}}|^2$ with $|p_{\sigma(j)}|=M_1$
on the left hand side is $C_{M_1, p_{\sigma(1)}}^2$ and the one on the right hand side is
$|b^{1}_{1; p_{\sigma(1)}}|^2+\dots+|b^{n_1}_{1; p_{\sigma(1)}}|^2$,
the vector $(1/C_{M_1,p_{\sigma(1)}})b_{1; p_{\sigma(1)}} \in \mathbb{C}^{n_1}$
is a unit vector.
On the other hand,
picking up the terms $(w_{1})^{p_{\sigma(1)}}(\ol{w_1})^{\hat{p}_{\sigma(1)}}$ with
$p_{\sigma(1)} \ne \hat{p}_{\sigma(1)}$ from the both sides of (\ref{M1}),
we have $<b_{1; p_{\sigma(1)}}, b_{1; \hat{p}_{\sigma(1)}}>=0$ for
$p_{\sigma(1)} \ne \hat{p}_{\sigma(1)}$.
Thus the set $\{(1/C_{M_1,p_{\sigma(1)}})b_{1; p_{\sigma(1)}}\}$
consisting of $T_{M_1, m_{\sigma(1)}}$ vectors is an orthogonal systems in
$\mathbb{C}^{n_1}$; and hence,
$T_{M_1, m_{\sigma(1)}} \leq n_1$.
This is the inequality required in (a).
For simplicity, we now denote by
$\{U_1, \dots, U_{T_{M_1, m_{\sigma(1)}}}\}$ this orthogonal system.
Then one can find unit vectors
$D_1, \dots, \\ D_{n_1-T_{M_1, m_{\sigma(1)}}} \in \mathbb{C}^{n_1}$
in such a way that
\begin{equation}
U=
\begin{pmatrix}
U_1 \\
\vdots \\
U_{T_{M_1, m_{\sigma(1)}}} \\
D_1 \\
\vdots \\
D_{n_1-T_{M_1, m_{\sigma(1)}}}
\end{pmatrix} \nn
\end{equation}
is an $n_1 \times n_1$ unitary matrix.
Note that the transformation
\begin{equation}
\Hn \ni (\wt{z}, \wt{w}_1, \dots, \wt{w}_N) \mapsto
(\wt{z}, \wt{w}_1\ol{U}^t, \wt{w}_2, \dots, \wt{w}_N) \in \Hn  \nn
\end{equation}
is an automorphism of $\Hn$.
Applying $U$ to $G_1$ as follows, we obtain the second part of the lemma.
\begin{align}
G_1\ol{U}^t
&=\Bigl(\sum_{|p_{\sigma(1)}|=M_1}b^1_{1;p_{\sigma(1)}}(w_{\sigma(1)})^{p_{\sigma(1)}},
   \dots, \sum_{|p_{\sigma(1)}|=M_1}b^{n_1}_{1;p_{\sigma(1)}}
        (w_{\sigma(1)})^{p_{\sigma(1)}} \Bigr) \ol{U}^t  \nn \\
&=(\dots, \dfrac{C^{2}_{M_1, m_{\sigma(1)}}}{C_{M_1, p_{\sigma(1)}}}(w_{\sigma(1)})^{p_{\sigma(1)}},
     \dots, 0, \dots) \nn \\
&=(\dots, C_{M_1, p_{\sigma(1)}}(w_{\sigma(1)})^{p_{\sigma(1)}}, \dots, 0, \dots). \nn
\end{align}
\end{proof}

The normalization of $G_1$ in Lemma~{\ref{dangelonormal}} is obtained by
restricting the situation to $w_{\sigma(2)}=\dots=w_{\sigma(N-1)}=w_N=0$.
By the same reason, restricting the situation to the variety
\begin{equation}
\Bigl(\bigcap_{\substack{j=1, \dots, N-1 \\ j \ne k}}\{w_{\sigma(j)}=0\} \Bigr) \cap \{w_N=0\},  \nn
\end{equation}
we can normalize $G_k$ as
\begin{equation}
G_k(w_{\sigma(k)})=(H_{M_k}(w_{\sigma(k)}), 0). \nn
\end{equation}

Restricting the mapping to $w_{\sigma(1)}=\dots=w_{\sigma(N-1)}=0$
and tracing the same argument as above, we can normalize $G^{\lambda}_{N}$
by using $T_{1, m_N}$ and $C_{1, p_N}$.
Then we conclude that $G_N$ is equivalent to $(w_N, 0)$.

\section{proof of the main theorem}
Now we have proved that $(F, G)$ is equivalent to
\begin{equation}
(z, H_{M_1}(w_{\sigma(1)}), 0, \dots, H_{M_{N-1}}(w_{\sigma(N-1)}), 0, w_N, 0)
\end{equation}
as a mapping between $\Hm$ and $\Hn$.
We pull back this normalization to the mapping between
$\Em$ and $\En$ via $\Psi$ and $\wt{\Psi}$.
By calculation, we obtain that
$(\mathcal{F}, \mathcal{G})$ is equivalent to
\begin{equation}
(z, H_{M_1}(w_{\sigma(1)}), 0, \dots, H_{M_{N-1}}(w_{\sigma(N-1)}), 0, w_N, 0). \nn
\end{equation}
This finishes the proof of the Main Theorem.

\section{Comparison with the previous result}
In this section, we compare our result with author's previous result in \cite{Ha2}.
Notation is the same as in the present paper.
\begin{mtTJM}
\itshape{Let $E(m;(m);(\alpha))$ and $E(n;(n);(\beta))$ be
generalized pseudoellipsoids with $N$ blocks.
Suppose that $2<m_j, 3<n_j$ and that
$n-m<\mathrm{min}\{n_1, \dots, n_N\}$.
Let $(\mathcal{F}, \mathcal{G}_1, \dots, \mathcal{G}_N) :
E(m;(m);(\alpha)) \to E(n;(n);(\beta))$
be a proper holomorphic mapping that is
holomorphic up to the boundary.
Assume that non-zero columns of any block row in the Jacobian matrix of
the unbounded representation
of $(\mathcal{F}, \mathcal{G})$ are linearly independent.
Then we have the following:

$\mathrm{(1)}$ There exists a permutation $\sigma$ of $\{1, \dots, N\}$ such that
$\mathcal{G}_j|_{w_{\sigma(j)}=0}=0$ for every $j$.

$\mathrm{(2)}$ If the permutation $\sigma$ in $\mathrm{(1)}$ satisfies the inequality
$m_{\sigma(j)} \leq n_j<2m_{\sigma(j)}-1$ for every $j$,
then $\alpha_{\sigma(j)}=\beta_j$ for all $j$ and the proper holomorphic mapping
$(\mathcal{F}, \mathcal{G}_1, \dots, \mathcal{G}_N)$
is equivalent to
$(\widetilde{\mathcal{F}}, \widetilde{\mathcal{G}}_1, \dots, \widetilde{\mathcal{G}}_N)$
of the form
\begin{equation}
\widetilde{\mathcal{F}}(z, w_1, \dots, w_N)=z, \;
\widetilde{\mathcal{G}}_j(z, w_1, \dots, w_N)=(w_{\sigma(j)}, 0), \;
1 \leq j \leq N. \nn
\end{equation}}
\end{mtTJM}
In \cite{Ha2}, we assume that
$m_{\sigma(j)} \leq n_j<2m_{\sigma(j)}-1$ for every $j$.
If we add this inequality to Main Theorem~{\ref{main}} as an assumption,
then, combining with (b) in Main Theorem~{\ref{main}},
we have an inequality:
\begin{equation}
\dfrac{(M_j+m_{\sigma(j)}-1)!}{M_j ! (m_{\sigma(j)}-1)!} \leq n_j < 2m_{\sigma(j)}-1. \label{dimineq}
\end{equation}
Note that
$\dfrac{(M_j+m_{\sigma(j)}-1)!}{M_j ! (m_{\sigma(j)}-1)!}$
is an increasing function with respect to $M_j$ and inequality (\ref{dimineq}) does not hold
for $M_j=2$.
Hence inequality (\ref{dimineq}) implies that $M_j=1$.
This implies $\alpha_{\sigma(j)}=\beta_j$ in Main Theorem~{\ref{main}},
which is the same conclusion as
in Main Theorem in \cite{Ha2}.
Since the function defined by (\ref{h}) satisfies $H_1(z)=z$,
under the assumption on dimensions
$m_{\sigma(j)} \leq n_j < 2m_{\sigma(j)}-1$ as in \cite{Ha2},
our mapping $(\mathcal{F}, \mathcal{G})$ is equivalent to the mapping
\begin{equation}
(z, w) \mapsto (z, w_{\sigma(1)}, 0, \dots, w_{\sigma(N-1)}, 0, w_N, 0), \label{totalembed}
\end{equation}
which
is also the same conclusion in \cite{Ha2}.
As a result, we conclude the following.
\begin{theorem}
We add the assumption $m_{\sigma(j)} \leq n_j < 2m_{\sigma(j)}-1$ to
Main Theorem~$\mathrm{\ref{main}}$.
Then it implies Main Theorem in \cite{Ha2}.
\end{theorem}
Since $M_j$ is invariant under the composition of automorphisms of $\Em$ and $\En$,
the mapping
\begin{equation}
(z, w) \mapsto
(z, H_{M_1}(w_{\sigma(1)}), 0, \dots, H_{M_{N-1}}(w_{\sigma(N-1)}), 0, w_N, 0) \nn
\end{equation}
with $M_j \not=1$ is not equivalent to the mapping (\ref{totalembed}).
Finally we would like to refer to the assumption on the Jacobian matrix in
Main Theorem in \cite{Ha2}.
Let $\mathcal{G}_j$ be a component of a proper holomorphic mapping as in
Main Theorem~${\ref{main}}$.
The assumption on the Jacobian matrix in Main Theorem in \cite{Ha2} means that
the non-zero columns of the Jacobian matrix of $\mathcal{G}_j$
are linearly independent for any $j=1, \dots, N-1$.
This condition is satisfied for $\mathcal{G}_j(w_{\sigma(j)})=(H_{M_j}(w_{\sigma(j)}), 0)$.

redpajama_set_name: RedPajamaArXiv


 Alfred - Meaning of Alfred, What does Alfred mean?
[ 2 syll. al-fred, alf-r-ed ] The baby boy name Alfred is pronounced as AEL-FRahD in English, AAL-FRahT in Dutch, aeL-FREHD in French, or AAL-FRehT in German and Polish †. Alfred is largely used in the Dutch, English, French, German, Polish, and Scandinavian languages, and it is derived from Old English origins. The name's meaning is inspired advice; old peace. A two-element name, it is from the elements 'aelf' meaning elf, supernatural being ; 'raed' counsel, advice. The name could be derived from Aelfraed (Old English), meaning 'inspired advice', or Ealdfrith (Old English) or Alfrid, meaning 'old peace'.
It was rather common before the Norman Conquest, and was borne by Alfred the Great (849-899), King of Wesse; he was widely respected for his inspiring rule, for driving the Danes out of England, and for his patronage of learning and culture. After the conquest, it was adopted in different forms by the Normans, and was one of the few Old English names that became popular beyond England and across the continent.
It later fell into disuse by the end of the Middle Ages before experiencing a revival among other medieval names in the 19th century. The form Alured was first coined in the 19th century, from a misreading of 'u' for the 'v' in the Latin form Alvred; 'v' was the transcription used for the alphabet 'f'. In literature, the name has appeared in the Charles Dickens novel Pickwick Papers (1836-1837) as Alfred Jingle. The form Alfie was popularized in England by the fictional young boy named Alfie in childrens' books written by Shirley Hughes in the mid 20th century, and later by the character Alfie Moon in the television soap opera East Enders (1985-). The name Alfreda (English, German, Italian, and Polish) is the female equivalent of Alfred.
The name Alfred is used to a great extent; it has 34 variants that are used in English and other languages. Variants used in English include Aelfraed, Ahlfred, Ailfred, Ailfryd, Alfea, Alfee, Alfeo, Alferd, Alfey, Alford, Alfre, Alfredas, Alfredo (used in Italian, Portuguese, and Spanish as well), Alfredos, Alfredus, Alfrey, Alfrid, Alfried, Alfryd, Elfrid, Freddie, and Freddy. Other English variants include the short forms Al, Alf, and Fred (used in German as well), and the diminutive forms Alfie, Alfy, and Allie.
Foreign variants of Alfred include Ailfrid (Irish), Alepeleke (Hawaiian), Alfréd (Hungarian), Alured (Latin and Latinized), and Fredo (Italian, Portuguese, and Spanish). Another foreign variant is the short form Fredi (German).
See also the related forms, Avery (English) and Avery (English).
See also the related categories, king, moon, medieval, supernatural (holy), norman, respected (honorable), young (cub), revival, peace (quiet), elf, polish, dutch, french, german, literature, opera, and scandinavian.
Alfred is a classic favorite. Currently it is still somewhat popular as a baby name for boys, though to a lesser extent than it was previously. At the height of its usage in 1928, 0.546% of baby boys were given the name Alfred. It ranked at #33 then. The baby name has since substantially fallen in popularity, and is of occasional use in recent years. In 2017, its usage was only 0.013% and its ranking #861. Within the group of boy names directly linked to Alfred, Avery was the most popular. In 2017, it was used 9 times more than Alfred.
Baby names that sound like Alfred include Ahlfred, Albert, Alberte, Albirt, Albrecht, Albret, Alburt, Albyrt, Alford, Alfrid, Alfried, Alfryd, Allford, Ulbrecht, Ulfred, Alberto, Alfredo, Alpertti, Ulfreid, and Ulfrid.
† Pronunciation for Alfred: AE as in "at (AE.T)" ; L as in "lay (L.EY)" ; F as in "fee (F.IY)" ; R as in "race (R.EY.S)" ; AH as in "mud (M.AH.D)" ; D as in "day (D.EY)" ; AA as in "odd (AA.D)" ; T as in "tee (T.IY)" ; EH as in "ebb (EH.B)"
Which Famous Persons bear the name Alfred?
Activist Al Sharpton, born Alfred Charles Sharpton, Jr., 3 October 1954, Brooklyn, New York.
Actor Alfred (Alfie) Bass, born Abraham Basalinksy, 8 April 1921 - 15 July 1987, Bethnal Green, London, England.
Author Damon Runyon, born Alfred Damon Runyan, 4 October 1880 - 10 December 1946, Manhattan, Kansas.
Cartoon Character Alfred E. Neuman, New York, New York.
Comedian Benny Hill, born Alfred Hawthorn Hill, Southampton, England.
Filmmaker Alfred Hitchcock, born 13 August 1899 - 28 April 1980, London, England.
Footballer Alf Ramsey, born Alfred Ernest Ramsey, 22 January 1920 - 28 April 1999, Dagenham, London, England.
Industrialist Alfred Nobel, born 21 October 1833 - 10 December 1896, Stockholm, Sweden.
Journalist Alistair Cooke, born Alfred Cooke, 20 November 1908 - 30 March 2004, Salford, Lancashire, England.
Mathematician Alfred North Whitehead, born 15 February 1861 - 30 December 1947, Ramsgate, Isle of Thanet, Kent, England.


'I turned myself into a pickle Morty!'. Crazed, space travelling granddad Rick, has been pushing the boundaries of science yet again, with this hilarious mug from Rick and Morty, featuring the caption 'I'm Pickle Rick'. Official merchandise.


You can talk the talk until you are blue in the face, but the application of what you're saying to your own life will demonstrate whether what you have to say is truly wise.
This month we continue to examine the description of wisdom James wrote. We will examine how Sow(ing) in Peace, will produce A Harvest of Righteousness.


text:
MP's fury as Cabinet pair warn against no-deal Brexit
CABINET duo Philip Hammond and Greg Clark infuriated Brexiteer Tory MPs yesterday by urging Britain's next prime minister to avoid a no-deal Brexit.
Business Secretary Mr Clark said the next Downing Street administration – expected to be led by Boris Johnson – should "strain every sinew" to avoid leaving the EU without a withdrawal agreement. He also claimed leaving the bloc without a deal would lead to "many thousands" of job losses. Chancellor Mr Hammond warned that a "constitutional crisis" would erupt if the next prime minister attempted to suspend Parliament to force through a no-deal Brexit.
But leading Eurosceptic Tory MP Andrew Bridgen lambasted them, saying: "We need to leave by October 31 and quite honestly, people like Greg Clark and Philip Hammond have been roadblocks to preparing for no deal."
Yesterday's spat between senior Tories follows Mr Johnson's promise to lead the UK out of the EU on October 31 "come what may, do or die".
Mr Clark warned over jobs in an interview on Sky News yesterday. He said: "It's evident that if you have the disruption that comes from a no-deal Brexit there will be people that will lose their jobs.
"It's many thousands of jobs. Everyone knows that."
In a separate interview, Mr Hammond told financial news service Bloomberg that any attempt to shut down Parliament "in order to carry out a course of action which Parliament is known to oppose" would provoke a constitutional crisis.
He added: "And, if we aren't able to prevent that course of action through Parliament, then, certainly, there will be resort to the courts, and I strongly support the position that Sir John Major has taken."
Former Tory prime minister Sir John has threatened legal action against any attempt to suspend Parliament in order to prevent MPs blocking a no-deal exit in October.
A spokeswoman for Theresa May said: "The Prime Minister has always been clear that leaving without a deal would be disruptive."

redpajama_set_name: RedPajamaCommonCrawl


WATCH and LISTEN. This story completes the adventure that began in Les trésors cachés. Students can read and listen along with this graphic novel as our two intrepid adventurers convince the principal to let them stay after school to track down the person responsible for stealing the students. Students will be surprised at who it is!.
White Board compatible CD including audio-visual format encourages group work and class discussion.


text:
Q: Android. Форматы экранов. Вот я делаю каждую картинку для 5 форматов (ldpi,mdpi и тд). При скачивании с гугл плея только одна папка в зависимости от телефона будет использоваться? Или же все остальные папки тоже мертвым грузом будут занимать место на телефоне?
A: Все ваши ресурсы будут лежать на телефоне после скачивания из Google Play. Хотите оптимизировать при первом запуске грузите из Web, определив параметры устройства.

redpajama_set_name: RedPajamaStackExchange


Dribbble is far from a new destination for inspiration, but their iOS app makes it easier than ever to see what others are designing and creating. You can browse projects, upload your own designs and even ask for tips from the community. It's the ultimate way to discover, connect and learn from world-class designers in every form of creativity that relates to screen printing, from posters to logo design.
Whether you're new to screen printing or a seasoned professional, it's worth checking out the design and screen printing courses available through the Skillshare app when you're on your next bus or train. Some courses are free and others are only available to Premium members, but the videos are clear, concise and range from designing repeating patterns to perfecting your halftones and textures at a more advanced screen printing level.
Tablet workspaces seem to be where the future is heading, and Adobe Illustrator has embraced this with their Touch option that makes the most of the Microsoft Surface and Windows tablets. It can take some practice to adapt from a mouse to a stylus, but the end result is that you're designing right there on the page wherever you happen to be. From here, you can simply treat your design as a standard Illustrator file to set up your screen print.
If you're screen printing for yourself or you are operating a screen printing business then effective accounting software can be invaluable in managing invoices, orders and inventory such as ink and blank T-shirts. Xero's mobile app lets you track your cash flow and record your expense claims so easily that you will always know exactly how your screen print business is tracking.


text:
Douglas County School
MLO/Bond Updates
District Boundary Maps
The Foundation for DCS
Cabinet Leadership
Apple Awards
Employee Rock Stars
Employee Appreciation Week
DCSD Strategic Plan
Strategic Themes
Phase I Initiatives
Schools & Academics
School Locator Map
Transition Bridge Program
Curriculum, Instruction, and Assessment
Educational Equity
Elementary Progress Report
Before and After School Enterprise (BASE)
DC Student Support Center
District Library Media Center
Early Childhood Education/Child Find
Literacy Interventions
Stone Canyon Outdoor EdVentures
Bus Pass (SMART Tag)
DCSD App
Dyslexia Information
Homeless Students/McKinney-Vento Services
Online Payments (MySchoolBucks)
Parent Portal (EngagED)
Parent Portal FAQ
Parent Portal (Infinite Campus)
School Accountability Committees
Student Data and Information Services
Weather Delays and Closures
Dating Violence Prevention
Safe2Tell Anonymous Reporting
Social Emotional Wellness Resources
Aesop (Substitutes)
Aesop (Teachers & Campus Users)
Child Abuse Reporting Form
Curriculum, Instruction and Professional Growth
District Library Media Center (DLMC)
DCSD Remote Learning Resources
Employee Council
Employee Dashboard
Learning Academy
Professional Development - Support Staff
Transportation Employee Portal
Employee Resources (continued)
Workday (Current Employees)
Workday (Pre Hires)
Student Data Privacy School Resources
Licensed Substitutes
Return to Careers Page
Covid Protocols for 2021-22 SY
Adjustments to COVID Outbreak/Exposure Notifications
Honoring Dr. Martin Luther King, Jr.
Returning From Winter Break - COVID Protocols
UPDATE: Face Coverings and Other COVID Mitigation Requirements
Plans for New Alt. Ed. School & Innovation Center
Honoring Our Veterans and Active Service Members
DCSD Board of Education
Court Decision Regarding Public Health Order
DCSD mask requirements for Monday
DCSD Update Regarding Its Masking Requirements
Clarification on Mask Procedures
Update: Mask Wearing and Quarantines
Update on DCSD Universal Mask Requirement
Mask Compliance for Visitors in Schools
Face Coverings Required for P-K12
Update - TCHD Public Health Order
New Public Health Order - Mask Requirements
Update on DCSD COVID Protocols for 2021-22
Welcome to the 2021-2022 School Year
Deputy Superintendents, SPED Services Officer
Douglas County School District » About » Leadership » Superintendent » Superintendent Messages » Covid Protocols for 2021-22 SY
Español | русский | 中文
Dear DCSD Families and Staff,
I hope you are enjoying your summer and some relaxation with family and friends! I am looking forward to the 2021-2022 school year and am excited to see our students and staff in person.
As we shared earlier this summer, DCSD is planning to return to as normal as possible learning environments this school year (more details below). DCSD will also welcome back parents and visitors into our schools this year! I am thrilled to kick off the new school year and look forward to our community staying healthy and well.
SOCIAL-EMOTIONAL LEARNING (SEL)
The COVID pandemic impacted all of us in many different ways. Our students may be experiencing heightened anxiety and trauma as they transition back to a more stable learning environment and reconnect with peers and educators. Our families and staff are also recovering from the disruption to their lives that was caused by the COVID pandemic which may result in stress, anxiety, uncertainty, and feeling overwhelmed. Our DCSD team will prioritize the social-emotional and mental health needs of all students and staff. We want to help our students transition out of a pandemic and into a successful school year. The beginning of the school year is a time to connect and build students into learning, activities, and athletics, while creating pride in their school experience. This connection is more critical than ever, and we have a commitment to build belonging and a systemic approach to monitor and support each and every student.
For example, kindergarten students will take part in the Balanced Beginnings program. This allows our young students to slowly transition into the classroom, while meeting other students and their teacher. Additionally, some middle and high schools have dedicated days for new students to visit the school, find their classroom and meet new friends. Your child's school will provide more details about these opportunities.
COVID PROTOCOLS FOR 2021-2022
In compliance with DCSD Board of Education policies, and ongoing collaboration with our partners at the Tri-County Health Department (TCHD) and the Colorado Department of Public Health and Environment (CDPHE), here is what the Douglas County School District is planning for the 2021-2022 school year.
Our COVID health and safety protocols will be in alignment with guidance from state and local health agencies. We recognize that the Centers for Disease Control (CDC) and American Academy of Pediatrics (AAP) recently released guidance highly encouraging students over the age of two to wear face coverings at school. These guidelines have been carefully considered in our collaborative decision making with the Tri-County Health Department. However, in light of strong vaccination rates in our state and county (71% in Douglas County to date), we feel confident moving forward with the following protocols. We will work with families to accommodate their students' needs and comfort levels so that every student has an enjoyable and safe learning environment.
Strong plans are able to adjust to changes while providing a framework that is defined, yet flexible. We will continue to monitor COVID data throughout the year and will adjust as needed should transmission rates increase in our schools, and/or public health orders change.
For students ages 11 and under: in alignment with local and state guidance, face coverings will be strongly encouraged while learning indoors, but will not be required.
For students ages 12 and over: in alignment with local and state guidance, face coverings will be strongly encouraged for anyone who is 12+ years and not fully vaccinated, but will not be required.
We will honor the decision of parents when it comes to their children wearing or not wearing a face covering in school. We will use our space in schools, classrooms and other areas to support each and every student with safe learning environments. Behavior such as "mask shaming" will not be tolerated.
CDPHE guidance states: The state recommends local public health agencies and school districts consider mask requirements for unvaccinated individuals, particularly in higher-risk environments. In addition, even when not required by local public health or a school district, unvaccinated and vaccinated staff and students may choose to wear masks. Schools and school districts should ensure that every classroom is a welcome environment for students and staff who choose to protect themselves in this manner.
Students or staff members who test positive for COVID will be required to isolate via past COVID practices.
Isolated students will be provided with learning opportunities until they are able to return to the classroom.
Our partners at Tri-County Health Department will continue contact tracing positive cases and advise on next steps when an exposure occurs.
CDPHE guidance states: Colorado will move from an individual case-based response strategy to a transmission mitigation strategy, where instead of quarantining students and staff after individual exposures, the risk of the whole school community, including the risk of interruptions to learning, is considered. Because in-person learning is essential for communities and students to thrive, CDPHE is recommending reduced incidents of quarantine in schools and child care specifically.
Schools will continue to implement physical distancing where possible, and make available additional physical space for those who want/need more distance while learning.
Disinfection/Ventilation
DCSD will continue with disinfection and ventilation protocols as follows:
Touchpoints (i.e. doorknobs, light switches, etc.) will continue to be disinfected throughout all buildings on a daily basis.
School buses will continue to be disinfected twice daily.
Teachers and staff will continue to have access to disinfectant supplies, including disinfecting spray, to clean high-touch areas regularly if desired.
The circulation of outdoor air within district buildings, via mechanical systems, will continue to be increased.
COVID Vaccination
The COVID vaccination is free to those 12+ years. Families may visit our website for location and registration information. We will continue to provide information to our community about COVID immunization opportunities as a courtesy for those who wish to access the vaccine. The COVID vaccine is not required for staff or students, however it creates more predictability and stability in our schools, activities and athletics. The higher the percentage of vaccinated students and staff, the better we can prepare for the school year and reduce the need for isolations. See current vaccination data for Douglas County.
In addition to the protocols listed above, DCSD will continue to encourage families to keep students at home when showing symptoms of illness.
I am eager to begin the new school year. One of my favorite sounds is that of a busy school, filled with students who are excited to learn!
Proud to be your superintendent!
Corey Wise
303-387-0100, 24hr Emergency 303-387-9999
In compliance with Titles VI & VII of the Civil Rights Act of 1964, Title IX of the Education Amendments of 1972, Section 504 of the Rehabilitation Act of 1973, the Age Discrimination in Employment Act of 1967, the Americans with Disabilities Act, the Genetic Information Nondiscrimination Act of 2008, and Colorado law, the Douglas County School District RE-1 does not unlawfully discriminate against otherwise qualified students, employees, applicants for employment, or members of the public on the basis of disability, race, creed, color, sex, sexual orientation, marital status, national origin, religion, ancestry, or need for special education services. Discrimination against employees and applicants for employment based on age, genetic information, and conditions related to pregnancy or childbirth is also prohibited in accordance with state and/or federal law. Complaint procedures have been established for students, parents, employees, and members of the public. The School District's Compliance Officer and Title IX Coordinator to address complaints alleging sexual harassment under Title IX is Aaron Henderson, 620 Wilcox Street, Castle Rock, Colorado, complianceofficer@dcsdk12.org, 720-433-1083.
Outside Agencies
Complaints regarding violations of Title VI, (race, national origin), Title IX (sex, gender), Section 504/ADA (handicap or disability), may be filed directly with the Office for Civil Rights, U.S. Department of Education, 1244 North Speer Blvd., Suite 310, Denver, CO 80204. Complaints regarding violations of Title VII (employment) and the ADEA (prohibiting age discrimination in employment) may be filed directly with the Federal Office of Equal Employment Opportunity Commission, 303 E. 17th Ave., Suite 510, Denver, CO 80202, or the Colorado Civil Rights Commission, 1560 Broadway, Suite 1050, Denver, CO 80202.


text:
Q: How to transfer data from a list to another list I'm trying to build a mission system which randomly pick 5 element from a list that store all the missions. But it seems I can't transfer the object from one list to another. Here is what I got so far.
void Awake()
{
    Object[] getScriptableObject = Resources.LoadAll("Scriptable Objects/Missions/sideMission");
    temp = new List<ScriptableObject>();
    for (int i = 0; i < getScriptableObject.Length; i++)
    {
        temp.Add((ScriptableObject)getScriptableObject[i]);
    }

    if (missionContainer.mission.Count < 5)
    {
        missionContainer.mission.Add(temp[Random.Range(0, temp.Count)]);
    }
    else
        return;
}

the temp List search every mission scriptable object in the folder and store them in the temp list, and I want the mission List to randomly pick 5 from the temp list and store them. What should I do?
 public class missionContainer : ScriptableObject
{
    public List<ScriptableObject> mission = new List<ScriptableObject>();
}

This is the mission list.

A: That would never add more than one mission. Don't you mean:
while (missionContainer.mission.Count < 5)
{
    missionContainer.mission.Add(temp[Random.Range(0, temp.Count)]);
}

But you are not removing the mission from the temp list, so you might add it again.

A: Actually you could use the generic typed version
// Or even better use the actual type of your ScriptableObject ;)
ScriptableObject[] getScriptableObject = Resources.LoadAll<ScriptableObject>("Scriptable Objects/Missions/sideMission");

and then simply do
temp = new List<ScriptableObject>(getScriptableObject);

And then it sounds like you rather wanted to either allow duplicate entries and use e.g.
// Note though that you could also simply directly use
// "getScriptableObject" array instead of "temp" in this case
while (missionContainer.mission.Count < 5)
{
    missionContainer.mission.Add(temp[Random.Range(0, temp.Count)]);
}

or if you need unique entries do something like e.g.
while (missionContainer.mission.Count < 5)
{
    if(temp.Count == 0)
    {
        Debug.LogWarning("Didn't find enough resources!", this);
        break;
    }

    var random = Random.Range(0, temp.Count);
    var item = temp[random];
    missionContainer.mission.Add(item);
    temp.Remove(item);
}

In case the missionContainer is actually also a ScriptableObject you might want to first call
missionContainer.mission.Clear();

in order to start with an empty list.


Victoria Wells is a dedicated musician teaching private in-home piano, voice, and organ lessons in the Washington, D.C. area.
Victoria Wells has been studying piano since she was five years old, and will graduate in May 2019 with her Bachelor of Arts in Piano Performance from Stetson University in DeLand, Florida. During her undergraduate studies, she was instructed by concert pianists Dr. Michael Rickman and Sean Kennard.
Victoria is an experienced classical pianist and vocalist with eight years of experience teaching piano and three coaching college level vocalists. She has played and sung with professional choirs such as Stetson University's Concert Choir and has collaborated with multiple opera and orchestra performances at Stetson University, including a production of Albert Herring by Benjamin Britten.
"To play a wrong note is insignificant; to play without passion is inexcusable." This quote by Ludwig van Beethoven embodies Victoria's lifelong philosophy regarding her own musicianship and the musicianship she desires to pass on to her students. Victoria's favorite aspect of teaching is getting to know each student individually, in order to best teach to his or her unique learning style. Her students have ranged from five-year-olds to senior citizens, with a variety of backgrounds in music from beginning to experienced classical musicians. Victoria can teach in a variety of styles including classical, gospel/hymn playing, jazz, and playing by ear, and seeks out every opportunity she can to learn new styles and techniques to share with her students.With a minor in political science from Stetson University and a passion for improving the world, Victoria will be pursuing a future not only in music, but also in law and not-for-profit work.


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You are at: Home › Members and Chamber Bodies › Members
Ing. Martin Kolovratnik
Date of birth: in year 1975
Constituency: Pardubice
Elected on the slate of: ANO2011
Seznam činností
member from 26. 10. 2013 till 26. 10. 2017
Electoral Committee, member from 4. 12. 2013 till 26. 10. 2017
Electoral Committee, vice-chairman from 5. 12. 2013 till 6. 12. 2013
Committee on Economic Affairs, member from 13. 2. 2014 till 26. 10. 2017
Committee on Economic Affairs, vice-chairman from 26. 2. 2014 till 26. 10. 2017
Committee on Economic Affairs, chairman from 28. 6. 2017 till 26. 10. 2017
Subcommittee on Transportation, member from 26. 2. 2014 till 26. 10. 2017
Electoral Commission of the Chamber of Deputies, member from 21. 11. 2013 till 26. 10. 2017
Electoral Commission of the Chamber of Deputies, chairman from 21. 11. 2013 till 26. 10. 2017
Political Group
Political group ANO 2011, member from 30. 10. 2013 till 26. 10. 2017
Interparliamentary Friendship Group
Czech republic - Republic of India, Democratic Socialist Republic of Sri Lanka, member from 29. 4. 2015 till 26. 10. 2017
The Supervisory Board of the State Fund of Transport Infrastructure, member from 7. 5. 2014 till 7. 5. 2018
What I have said
How I voted
Volba volebního období
Other Election Term
Chamber of Deputies (from 2017)


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Home State Laws	Shelby County v. Holder: the state of voting rights 6 years later
Shelby County v. Holder: the state of voting rights 6 years later
written by Ex Offenders Vote June 25, 2019
In recent years, threats to voting rights and the possibility of voter suppression have become increasingly serious concerns for civil rights groups. This was on full display in the 2018 midterm elections, as voting laws and regulations in several states impacted high-stakes races in ways that disproportionately affected voters of color.
But the concerns raised by these elections did not originate with high-profile 2018 contests in places like Georgia and Florida, both states where black, Latino, and Asian American voters struggled to cast a ballot. Instead, these issues can in part be directly traced back to a 2013 Supreme Court ruling in the case Shelby County v. Holder.
That ruling, which turns six years old this week, invalidated a key portion of the Voting Rights Act of 1965, long seen as one of the most important civil rights laws of the past century. On June 25, 2013, the court's conservative majority ruled that the government was using an outdated and unconstitutional process to determine which states were required to have their voting rules approved by the government. Before the ruling, nine states (and several other counties and townships) had been subjected to this requirement.
This "preclearance" was intended to ensure that these areas — each with a history of discriminatory voting laws — did not pass new restrictions impacting the ability of voters of color to cast a ballot. But a county in one such state, Alabama, argued that the requirements (all part of Sections 4 and 5 of the Voting Rights Act), were excessive and unnecessary nearly 50 years after the legislation's passage. When a majority of the court agreed, it dealt a heavy blow to what civil rights advocates have long seen as the "heart of the Voting Rights Act."
Members of the NAACP speak shortly after the Supreme Court invalidated a key portion of the Voting Rights Act of 1965 on June 25, 2013.
Pete Marovich/MCT via Getty Images
The ruling had an impact that continues today, with several civil rights and voting rights groups telling Vox that the post-Shelby era has presented various difficulties: an uptick in legal actions taken against states, increased costs for monitoring and pursuing litigation over voting restrictions, and, perhaps most significantly, more laws creating new requirements in the voting process — many of them disproportionately affecting black voters and other communities of color.
"Since Shelby, states have really opened the floodgates to voter suppression, and we've seen laws that have discriminated against voters of color all across the country," says Leigh Chapman, the director of the voting rights program at the Leadership Conference on Civil and Human Rights.
Advocates say that in a post-Shelby world, not only have they have lost a key tool in ensuring all voters have an equal and unimpeded right to cast a ballot, voting rights enforcement has also become much harder.
What the Shelby County v. Holder decision meant for the Voting Rights Act
When the Voting Rights Act was passed in 1965, it included provisions requiring that states with low black voter turnout and a history of discriminatory voting laws have future elections changes federally approved to ensure that they did not hamper voters from marginalized communities.
To do this, the VRA relied on two closely linked provisions. Section 5 of the act established that states could be subjected to "preclearance," requiring all major voting and election systems changes to be approved by the Justice Department or a federal court. To determine which areas could be placed under preclearance, the VRA relied on a different part of the legislation — known as Section 4b — which created a formula outlining the types of histories that needed to be present in a jurisdiction.
In 2010, officials representing Shelby County, Alabama — which at the time was under preclearance — filed a federal lawsuit against the Justice Department and then-Attorney General Eric Holder, challenging the constitutionality of both Section 4b and Section 5, arguing that they were outdated and should no longer be used. DC courts ruled against the county in 2011 and on its first appeal 2012, but county officials appealed to the Supreme Court.
In a 5-4 ruling, the court's conservative members argued that the conditions that justified preclearance in 1965 didn't apply in 2013. "Coverage today is based on decades-old data and eradicated practices," Chief Justice John Roberts wrote in the majority opinion. While racial disparities in voter registration and turnout "was compelling evidence justifying the preclearance remedy and the coverage formula" in 1965, Roberts said that "there is no longer such a disparity."
A group of Alabama residents stand outside of the Supreme Court's oral arguments in Shelby County v. Holder on February 27, 2013.
However, the Shelby ruling was slammed by the court's more liberal justices, who argued that the discrimination and racial disparities in voting that prompted the creation of preclearance had not disappeared. Justice Ruth Bader Ginsburg wrote in her dissent that "throwing out preclearance when it has worked and is continuing to work … is like throwing away your umbrella in a rainstorm because you are not getting wet."
The Shelby ruling released nine states — Alabama, Alaska, Arizona, Georgia, Louisiana, Mississippi, South Carolina, Texas, and Virginia — from the previous preclearance coverage formula. The ruling also affected some counties and townships in California, Florida, Michigan, New York, North Carolina, and South Dakota.
It's important to note that the ruling itself did not find Section 5 unconstitutional and instead focused solely on the coverage formula in Section 4b. To resume using Section 5, the court said that Congress, which created the VRA in 1965, needed to enact a different, constitutional preclearance requirement. But Congress, despite the efforts of Democrats, still has not done so, leaving little federal oversight of voting in the US.
Voting rights advocates say that the striking down of Section 4b — and the resultant declawing of Section 5 — blunted one of the strongest measures that held jurisdictions accountable for their proposed voting laws.
"Section 5 was a sunshine law," says Leah Aden, the deputy director of litigation for the NAACP Legal Defense and Educational Fund. "The burden was on jurisdictions to prove that a law wasn't going to make the experience of black and Latino voters, and Native American voters, and Asian American voters worse off."
Since Shelby, many states have passed controversial restrictive voting laws that impact communities of color
According to the Brennan Center for Justice, a policy and research group that monitors new voting restrictions, there have been hundreds of "harsh measures making it harder to vote" introduced in state legislatures since 2010. Many of these were introduced after the 2013 Shelby ruling, and, as a federal commission noted last year, have been seen both in states previously subjected to preclearance and states that were not.
These restrictions have taken many forms, including strict photo ID requirements, limitations on who can provide assistance at polling places, the curbing of early voting days, and the closing of hundreds of polling places across the US. Other measures, like the purging of voters from state voter rolls and drawing election districts in a way that curbs the power of voters of color, have affected how much power communities of color hold in elections.
"Shelby has made it harder — and made it more important — to affirmatively protect these communities from the spread of restrictions," John Yang, the president and executive director of Asian Americans Advancing Justice-AAJC, tells Vox.
Yang notes that laws and election management practices — some of which limit the availability of ballots in different languages, have strong photo ID restrictions, and scrutinize the ballots of voters who are naturalized citizens — have all negatively affected Asian American voters. Organizations working with other marginalized communities, including Latino voters, black voters, and Native American voters, have also noted that other racial groups are disproportionately affected by restrictions passed in the six years since Shelby.
The passing of these laws has been made possible by the defanging of preclearance, which allowed the federal government to weigh in on these restrictions before they could take effect, creating some standardization in what was not allowed in state election systems. After Shelby, though, states were allowed to enact measures that would likely have been reviewed by Section 5 in earlier years.
Civil rights groups say this is especially the case in places like Texas, where a strict voter ID requirement first passed in 2011 (the law did not take effect until 2013 when the state was no longer under preclearance) mandated that voters show a government-issued photo ID, driver's license, or passport before voting. That requirement was later altered by the state legislature in 2017 after a court order, but the decision was not supported by the federal government, with the Justice Department reversing its earlier opposition to the law shortly after President Donald Trump entered office.
A polling place in Austin, Texas prepares to open on November 6, 2018.
Suzanne Cordeiro/AFP/Getty Images
More recently, Texas officials were sued after a botched review of the state's voter rolls resulted in the state unnecessarily challenging the citizenship status of thousands of voters, many of whom are naturalized citizens. Acting Texas Secretary of State David Whitley, a Republican, later resigned from office after agreeing to end the controversial review as part of a settlement.
Thomas Saenz, president and general counsel for the Mexican American Legal Defense and Educational Fund (MALDEF), says that restrictions in places like Texas have been particularly hard for voters who immigrated to the US. He adds that many of the restrictions put in place in recent years have affected Latino voters, a large and growing portion of the state's electorate.
"The right to vote is essential to every other right given to communities of color and other populations," he says. Saenz adds that while the Shelby ruling has clearly had an impact on voters of color in multiple states, it won't be until the next round of redistricting after the 2020 Census — the first where former Section 5 states can draw maps without getting them pre-approved — where the "full scope of the loss caused by Shelby" will be seen.
Civil rights groups have had successes and losses in the fight since Shelby
But threats to Latino voters have led to at least one successful effort to restore preclearance in a part of Texas. In 2017, a lawsuit from MALDEF and other groups argued that election officials in Pasadena, Texas, had diluted the power of Latino voters when they created two at-large seats on Pasadena's eight-seat city council. To do so, the city condensed residents into six districts, a move that halted the momentum for districts with predominantly Latino voters to gain a majority of council seats. A federal judge ruled that Pasadena violated the Voting Rights Act, and placed the city under preclearance.
Other important victories for voting rights advocates came a year earlier in 2016, when a lawsuit over a set of strict North Carolina voting laws introduced immediately after the Shelby decision led to a federal court ruling that the state had targeted black voters with "almost surgical precision" in an effort to reduce their turnout. North Carolina Republicans later appealed the ruling in the hopes of restoring the law, but the Supreme Court declined to hear the case in 2017.
But not every high-profile voting restriction has been overturned by a court or blocked ahead of an election. Some restrictions, like a North Dakota law requiring photo IDs to have a residential address, including for those living on tribal reservations without formal streets, have been upheld even as lawyers argued that they disproportionately affected Native Americans' ability to vote.
And a series of lawsuits over several voting laws and restrictions in Georgia occurred in the background of the state's 2018 gubernatorial election. Voting rights advocates argued that Brian Kemp, then Georgia's secretary of state and a Republican gubernatorial candidate, had diluted the power of voters of color through strict voter registration requirements, the closure of more than 200 polling places, and the purging of hundreds of thousands of voters. Critics said Kemp also failed to adequately prepare for long lines and potential voting issues on Election Day.
Kemp won the election, but the state is currently facing a lawsuit from Fair Fight Action, a group founded by Kemp's Democratic gubernatorial opponent Stacey Abrams. Fair Fight argues that voting restrictions and other barriers in the state created a system that effectively disenfranchised many Georgia voters, especially voters of color. The lawsuit calls for Georgia to be placed back under federal preclearance.
The House Oversight Committee is also investigating allegations of voter suppression in Georgia, Texas, and Kansas.
Many of these lawsuits are possible under Section 2 of the Voting Rights Act, which allows for voting laws to be struck down if they are found by a court to be discriminatory. But voting rights advocates say that stopping restrictions in this way is far less efficient than the use of Section 5. This is largely due to the fact that instead of reviewing a restriction and stopping it before it even takes effect, Section 2 means that a law is usually enacted as a lawsuit over it continues. Voting rights groups say this requires more effort from civil rights groups to challenge a law, and also creates additional expenses both for civil rights groups and county and state officials.
"We contend that if Section 5 had been in place, we would not have seen the sort of mess that we've seen," says Aden of the NAACP. "Civil rights groups could be working on other things but instead we're using our resources to deal with things that Section 5 would've addressed more easily."
With these problems in mind, voting rights advocates say it's time to fully restore the Voting Rights Act
As restrictions in the wake of Shelby v. Holder have been found to disproportionately affect — and in some cases deliberately target — voters of color, lawmakers behind these restrictions have argued that laws are not discriminatory.
Instead, a primarily Republican group of lawmakers have said that the measures are needed to combat voter fraud, even though there's little evidence that significant fraud exists. These lawmakers also say that their efforts are a legally acceptable attempt to carve out a partisan advantage rather than deliberately curb the voting power of black, Latino, and Asian American voters, despite the fact that due to the demographic nature of the current political parties, these goals are often one and the same.
In the states once subject to preclearance, these issues would have been halted to some extent before Shelby. But without the full VRA in place, voters and civil rights groups have been forced into a largely reactionary stance, filing lawsuits over restrictions after they have been enacted or passed by legislatures rather than being able to get a federal review of them before their implementation.
Alabama Rep. Terri Sewell (center), speaks at a news conference introducing the Voting Rights Advancement Act on February 26, 2019.
Cheriss May/NurPhoto via Getty Images
It is also true that many states with a significant number of voting laws on the books are still looking to pass new restrictions. For example, at the same time that lawyers in Texas were in court preparing to settle the lawsuit over the unnecessary flagging of naturalized citizens, the state legislature was considering a measure that would create new criminal penalties for people who attempted to vote without knowing they were ineligible. The measure did not pass before the end of the 2019 legislative session.
And Tennessee, the 49th state in the country in voter turnout in 2016, recently enacted — and was subsequently sued over — a law that fines and punishes voter registration groups for submitting too many incomplete and inaccurate forms.
As these challenges remain, voting rights groups argue that it is necessary for Congress to restore the Voting Rights Act by enacting a new system for preclearance. Such a system is included in the Voting Rights Advancement Act, also known as HR 4, a measure introduced by Democrats earlier this year. The measure would "give the federal government a stronger ability to take action against states with a history of discrimination," according to Vox's Ella Nilsen.
"We can't unring the bell," the bill's sponsor, Alabama Rep. Terri Sewell, told Nilsen in December. "What we've seen is that since Shelby, more than 30 states have imposed greater requirements for voting, and in a lot of those states, we've seen elections take place that have later been found to have had intentional discrimination."
Sewell's measure would subject states to preclearance if they had a history of discrimination in its voting system or electoral maps in the past 25 years. The formula would rely on more recent examples of voting rights violations than Section 4b did before it was struck down.
Sewell's bill, which was initially part of a broader set of Democrat-sponsored anti-corruption reforms known as HR 1, the For the People Act, would also create a preclearance requirement for any voting or election law on a list of "known practices" — measures already shown to disproportionately affect voters of color. Chapman, the Leadership Conference voting rights director, says that this provision in particular "would really modernize the preclearance formula."
Other efforts to restore the Voting Rights Act have come from 2020 presidential candidates. Beto O'Rourke, for example, has a voting rights and democracy reform platform that calls for automatic voter registration, cracking down on voter ID laws, limiting aspects of voter roll purges, and also pushes against other voting restrictions. And Sen. Elizabeth Warren (D-MA) used the 2019 Shelby anniversary to announce her own voting rights platform that calls for the passage of both the Voting Rights Advancement Act and the Native American Voting Rights Act as well as several election security measures.
Chapman says that the attention candidates are giving voting rights matters in the larger push for momentum for additional changes to voting rights in the US, but she notes that restoring the VRA itself largely lies with Congress.
Still, as another Shelby County v. Holder anniversary arrives without full government oversight of the VRA, a protection that has now been missing for multiple election cycles and seems likely to remain absent in the 2020 elections, advocates say that it is imperative that voting rights be protected and that the country move toward expanding — not limiting — access to the ballot.
"Our voting experience in this country should not be that things are relatively easy for certain voters, while setting up hurdles for others," Aden says. "We should not have a two-tiered system of democracy."
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11.29.00 - "Children's Social and Emotional Development: Essential Building Blocks for Learning"
MEDIA ADVISORY : "Children's Social and Emotional Development: Essential Building Blocks for Learning"
"Children's Social and Emotional Development: Essential Building Blocks for Learning", a day-long workshop at the University of California, Berkeley, will present recent research on young children. Teachers and child care workers will hear from campus researchers how they can foster optimal conditions for learning and how they can interpret a child's troublesome behavior.
WHEN: Saturday, Dec. 2, from 9 a.m. to 3:30 p.m.
WHERE: Clark Kerr Conference Center/Krutch Theater, located at the corner of Waring Street and Parker Street, near UC Berkeley's main campus.
WHO: The Institute of Human Development at UC Berkeley is sponsoring its second annual workshop on learning and development in young children, aimed at applying the latest research to pre- and elementary school education in the Bay Area. Co-sponsors are the UC Berkeley Chancellor's Pledge Fund and the Alameda Co. Commisson for Prop. 10.
* A violence prevention program called Safe Start, implemented experimentally in 15 Bay Area child care centers, has led to a decline in aggression and withdrawn behavior among the children who participated.
* Children who are rejected or neglected by their peers develop behavioral and health problems that can be alleviated by teachers with specific interventions.
* How do researchers predict attention deficit hyperactivity disorder (ADHD) in children of preschool age who are already naturally impulsive and hyperactive?
* Teasing in children can have positive effects, helping them to negotiate conflict, socialize one another and express affection.
* Researchers can now trace the outlines of problematic behavior in preschool children and talk to parents about how to deal with these problems.


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Interested in participating in field trials?


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The fire is burning on a construction site near the Whole Foods and an Acura dealership.
People nearby are being evacuated.
Stay with ABC7 News on air and online for the latest details.